The terrestrial water cycle partitions precipitation
between its two ultimate fates: “green water” that is evaporated or
transpired back to the atmosphere, and “blue water” that is discharged to
stream channels. Measuring this partitioning is difficult, particularly on
seasonal timescales. End-member mixing analysis has been widely used to
quantify streamflow as a mixture of isotopically distinct sources, but
knowing where streamwater comes from is not the same as knowing where
precipitation goes, and this latter question is the one we seek to answer.
Here we introduce “end-member splitting analysis”, which uses isotopic
tracers and water flux measurements to quantify how isotopically distinct
inputs (such as summer vs. winter precipitation) are partitioned into
different ultimate outputs (such as evapotranspiration and summer vs. winter
streamflow). End-member splitting analysis has modest data requirements and
can potentially be applied in many different catchment settings. We
illustrate this data-driven, model-independent approach with publicly
available biweekly isotope time series from Hubbard Brook Watershed 3. A
marked seasonal shift in isotopic composition allows us to distinguish
rainy-season (April–November) and snowy-season (December–March)
precipitation and to trace their respective fates. End-member splitting
shows that about one-sixth (18±2 %) of rainy-season precipitation
is discharged during the snowy season, but this accounts for over half
(60±9 %) of snowy-season streamflow. By contrast, most (55±
13 %) snowy-season precipitation becomes streamflow during the rainy
season, where it accounts for 38±9 % of rainy-season streamflow.
Our analysis thus shows that significant fractions of each season's
streamflow originated as the other season's precipitation, implying
significant inter-seasonal water storage within the catchment as both
groundwater and snowpack. End-member splitting can also quantify how much of
each season's precipitation is eventually evapotranspired. At Watershed 3,
we find that only about half (44±8 %) of rainy-season precipitation
evapotranspires, but almost all (85±15 %) evapotranspiration
originates as rainy-season precipitation, implying that there is relatively
little inter-seasonal water storage supplying evapotranspiration. We show
how results from this new technique can be combined with young water
fractions (calculated from seasonal isotope cycles in precipitation and
streamflow) and new water fractions (calculated from correlations between
precipitation and streamflow isotope fluctuations) to infer how
precipitation is partitioned on multiple timescales. This proof-of-concept
study demonstrates that end-member mixing and splitting yield different, but
complementary, insights into catchment-scale partitioning of precipitation
into blue water and green water. It could thus help in gauging the
vulnerability of both water resources and terrestrial ecosystems to changes
in seasonal precipitation.
Introduction: end-member mixing and end-member splitting
End-member mixing analysis has been widely used in isotope hydrograph
separation, as well as in other applications that seek to interpret
environmental flows as mixtures of chemically or isotopically distinct
end-member sources (see Klaus and McDonnell, 2013, and references therein).
The simplest form of end-member mixing analysis uses a single conservative
tracer to estimate the fractions of two sources in a mixture (see Fig. 1).
It is derived from the mass balances for the water and tracer,
qA→M+qB→M=QM
and
qA→Mδ‾A+qB→Mδ‾B=QMδ‾M,
where qA→M and qB→M denote fluxes from
end-members A and B to a mixture M whose
total flux is QM, and the volume-weighted isotope signatures
(or tracer solute concentrations) in these three fluxes are δ‾A, δ‾B, and δ‾M, respectively. These equations embody the two essential
assumptions of end-member mixing analysis: that the mixture M is
sourced from (and only from) A and B (Eq. 1) and that
the tracer is conservative, with no other sources or sinks that alter the
tracer signatures δ‾A and δ‾B between the end-members A and B and
the mixture M (Eq. 2). Simultaneously solving Eqs. (1) and (2)
yields the well-known end-member mixing equations
fM←A=qA→MQM=δ‾M-δ‾Bδ‾A-δ‾BandfM←B=1-fA=qB→MQM=δ‾M-δ‾Aδ‾B-δ‾A,
where fM←A and fM←B denote
the fractions of the mixture M originating from the two sources
A and B. Using only tracer signatures, Eq. (3) can
determine the relative fractions of the two end-members in the mixture, even
if all of the relevant fluxes (qA→M, qB→M,
and QM) are unknown.
Schematic illustration of end-member mixing and end-member
splitting. Two end-members, A and B,
contribute to a mixed outflow M and to two other
outflows, denoted X and Y,
respectively. The fluxes between the end-members and outflows are denoted
qA→M, qA→X, qB→M, and
qB→Y; these are assumed to not be
directly measurable. Conventional end-member mixing, as shown at the bottom
of the figure, can be used to calculate the fractions of the two end-members
in the mixture using only their volume-weighted average tracer signatures
(δ‾A, δ‾B, and δ‾M). If one
also knows the water fluxes in the mixed outflow and one or both
end-members, one can use end-member splitting, as shown on the left- and
right-hand sides of the figure, to quantify how the end-members are partitioned
between the mixture M and their other outflows X and Y.
For many hydrological problems, it would be helpful to know not only how
end-members are combined in mixtures, but also how individual end-members
are partitioned among their possible fates (e.g., Welp et al., 2005). That
is, it would be helpful to know not only how end-members are mixed (as shown at
the bottom of Fig. 1), but also how they are split into different fluxes (as
shown on the left- and right-hand sides of Fig. 1). Whereas end-member mixing has been
widely explored in hydrology, the potential for new insights from end-member
splitting has been less widely appreciated. What fraction of winter precipitation
becomes winter streamflow? What fraction becomes summer streamflow? What
fraction eventually evaporates or transpires? Questions like these require
understanding how end-members (such as snowmelt in this example) are split
among their potential fates, rather than how they are mixed.
Recent work hints at the potential benefits of an end-member splitting
approach. von Freyberg et al. (2018b) have recently shown that one can gain
new insights into storm runoff generation by expressing the flux of event
water in the storm hydrograph (the classic subject of isotope hydrograph
separation) as a fraction of total precipitation rather than total
streamflow. In our terminology, von Freyberg et al. (2018b)'s approach splits storm
rainfall into two fractions: one that becomes “event water” during the
current storm and another that eventually either evapotranspires or is
stored in the catchment, to become base flow or “pre-event water” in future
hydrologic events. Similarly, Kirchner (2019a, Sects. 2.6, 2.7, 3.5, and 4.7)
has shown how tracer data can be used to estimate “forward new water
fractions” and “forward transit time distributions”, which quantify the fate
of current precipitation (rather than the origins of current streamflow,
which is the focus of most conventional approaches to transit time
estimation). These “forward” new water fractions and transit time
distributions quantify how current precipitation is split among future streamflows,
rather than quantifying how past precipitation events are mixed in current
streamflow. The underlying concept is not new, dating back at least to Eq. (7)
of Niemi (1977) in the context of transit time distributions. However, it
has not been widely recognized that a similar approach can also be applied
in end-member mixing analysis, to infer the partitioning of the end-members
themselves. Our purpose here is to outline the potential of this approach,
which we call end-member splitting.
End-member splitting is based on the observation that (for example) the
fraction of end-member A that becomes mixture M
(end-member splitting) is directly related to the fraction of mixture
M that is derived from end-member A (end-member
mixing). These fractions both have the same numerator, the flux
qA→M that flows from A to M; they
just have different denominators, QA in the first case and
QM in the second (see Fig. 1). This in turn implies that we
can perform end-member splitting by rescaling the results of end-member
mixing, through multiplying by the ratio of QM to
QA:
ηA→M=qA→MQA=QMQAqA→MQM=QMQAfM←A=QMQAδ‾M-δ‾Bδ‾A-δ‾B,
where ηA→M is the proportion of end-member A
that eventually becomes mixture M and fM←A is the fraction of mixture M that originated as end-member
A. Since all of end-member A must eventually become
either part of mixture M or another output (or combination of
outputs), here denoted X, we can straightforwardly calculate
ηA→X, the fraction of A that eventually
becomes X, by mass balance:
ηA→X=qA→XQA=1-ηA→M=1-QMQAδ‾M-δ‾Bδ‾A-δ‾B.
One can also directly calculate the magnitudes of the fluxes connecting each
end-member to each output, e.g.,
qA→M=QAηA→M=QMfA=QMδ‾M-δ‾Bδ‾A-δ‾B
and
qA→X=QAηA→X=QA-QMδ‾M-δ‾Bδ‾A-δ‾B.
We use the symbol η to represent how an end-member is partitioned
among multiple outputs, to explicitly distinguish it from the mixing
fraction f, which represents how a mixture is composed of multiple
end-members. We specifically use the symbol η because in
thermodynamics it represents efficiency, and ηA→M (for
example) can be interpreted as the efficiency with which end-member
A is transformed into the mixed output M.
If the unsampled outputs X and Y can be pooled together (for example, as
annual evapotranspiration fluxes), we can straightforwardly calculate the
fractional contributions of each end-member to this pooled output (here
denoted XY) as
fXY←A=qA→XYQXY=QAQXYηA→X=QAQXY1-ηA→M=QA-QMδ‾M-δ‾Bδ‾A-δ‾BQA+QB-QM.
This calculation requires not only that the fluxes QA,
QB, and QM are known, but also that they are known
precisely enough that the mass balance
QXY=QA+QB-QM can
be quantified with reasonable accuracy.
Whereas end-member mixing only requires measurements of the volume-weighted
tracer composition in the mixture and all of its potential sources,
end-member splitting additionally requires measurements of the water fluxes
in the end-members and mixture(s). Both end-member mixing and end-member
splitting analyses should always be accompanied by uncertainty estimates
(quantified via, for example, Gaussian error propagation), to avoid
over-interpretation of highly uncertain results. Gaussian error propagation
formulas for the main equations in this paper are presented in the
Supplement, and quantities in the main text and the figures are shown
± standard errors.
Like end-member mixing, end-member splitting can be generalized to more than
two sources if the number of tracers equals at least the number of sources
minus one, and if the tracers are sufficiently uncorrelated with one
another. End-member splitting can also be generalized straightforwardly to
any number of mixtures, even using only one tracer if each mixture combines
only two end-members; in the general case, the number of
(not-too-correlated) tracers in each mixture must equal at least the number
of end-members minus one.
Proof-of-concept applicationField site and data
As a proof-of-concept demonstration, here we apply end-member splitting
analysis to Campbell and Green's (2019) measurements of δ18O
and δ2H at Hubbard Brook Experimental Forest, Watershed 3.
Campbell and Green (2019) measured δ18O and δ2H in
time-integrated bulk precipitation samples, and instantaneous streamwater
grab samples, taken at Watershed 3 approximately every 2 weeks between
October 2006 and June 2010 (Fig. 2); the isotope sampling and analysis
procedures are documented in Green et al. (2015). We also used daily
precipitation and streamflow measurements for Watershed 3 compiled from 1958
through 2014 by the USDA Forest Service Northern Research Station (2016a,
b).
(a) Time series of daily water fluxes and biweekly deuterium
values in streamwater (dark blue) and precipitation (light blue) at
Watershed 3, Hubbard Brook Experimental Forest (data of Campbell and Green,
2019). (b) Dual-isotope plot showing the local meteoric water line computed by
volume-weighted regression (δ2H=4.74±2.26+ (7.37±0.22) δ18O). Streamwater lies slightly above the
local meteoric water line, on average (lc-excess =2.91±0.27 ‰, mean ± standard error), possibly suggesting
slight evaporative fractionation of precipitation within the sample
collector or potential fractionation of streamwater by sub-canopy moisture
recycling (Green et al., 2015).
Watershed 3 is a small (42.4 ha) headwater basin that has served as a
hydrologic reference watershed for manipulation experiments conducted in
several other nearby watersheds (Bailey et al., 2003). Its soils are
well-drained Spodosols with a 3–15 cm thick, highly permeable organic layer
at the surface, underlain by glacial drift of highly variable thickness
(averaging roughly 0.5 m, Bailey et al., 2014), which in turn overlies
schist and granulite bedrock that is believed to be highly impermeable
(Likens, 2013). Ground cover is northern hardwood forest, comprising mainly
American beech (Fagus grandifolia Ehrh.), sugar maple (Acer saccharum
Marsh.), and yellow birch (Betula alleghaniensis Britt.) (Green et al.,
2015), with a growing season extending from June through September (Fahey et
al., 2005). Watershed 3 has a humid continental climate, with average
monthly temperatures ranging from -8∘C in January to 18 ∘C in July (Bailey
et al., 2003). Annual average precipitation was 136 cm yr-1 from
1958 through 2014, distributed relatively evenly throughout the year, and
annual average streamflow was about 87 cm yr-1, implying
evapotranspiration losses of roughly 49 cm yr-1, or about one-third
of average precipitation (USDA Forest Service Northern Research Station,
2016a, b). Approximately 30 % of annual precipitation falls as snow,
mostly from December through March, reaching an average annual maximum
accumulation of 19 cm snow water equivalent (Campbell et al., 2010) and
supplying springtime snowmelt pulses in streamflow, which typically peak in
April.
We adjusted Campbell and Green's precipitation isotope values to account for
the difference between the mean catchment elevation (642 m; Ali et al.,
2015) and the elevation at the precipitation sampler (564 m; Campbell and
Green, 2019) assuming an isotopic lapse rate of -0.28 ‰ per 100 m for δ18O (Poague and
Chamberlain, 2001) and 8 times this amount (-2.24 ‰ per 100 m) for δ2H. We weighted each
precipitation isotope value by the cumulative precipitation that fell during
each sampling interval to calculate seasonal volume-weighted averages of
δ18O and δ2H in precipitation. To calculate
seasonal volume-weighted averages of δ18O and δ2H
in streamflow, we weighted each streamflow isotope value by the cumulative
streamflow since the previous sample. We calculated uncertainties for all
derived quantities using Gaussian error propagation, based on the standard
errors of the average water fluxes and the volume-weighted standard errors
of the average isotope ratios, as described in the Supplement. An R script that performs the end-member mixing and splitting calculations, along with the accompanying error propagation, is available online (Kirchner, 2019b). Quantities
are reported ± standard errors.
Isotope signatures in Hubbard Brook precipitation exhibit the typical
seasonal pattern of temperate mid-latitudes (Fig. 2a): precipitation is
isotopically lighter during winter and heavier during summer. There is also
considerable sample-to-sample variability, presumably reflecting differences
in water sources, atmospheric moisture trajectories, and atmospheric
dynamics between individual precipitation events. The streamwater samples
lie slightly above the local meteoric water line (Fig. 2b), suggesting that
either the precipitation samples have been slightly affected by evaporative
fractionation within the sample collector or that the streamwater samples
have been affected by sub-canopy moisture recycling (Green et al., 2015).
The seasonal cycle in precipitation isotopes is preserved in streamwater at
Watershed 3 (somewhat damped and phase-shifted), whereas the shorter-term
fluctuations in precipitation isotopes are almost entirely damped away (Fig. 2a). The strong damping in short-term isotope fluctuations indicates that
“event” water from recent precipitation comprises only a small fraction of
streamflow, which instead consists mostly of “pre-event” water from many
previous precipitation events, thus averaging together their isotopic
signatures (Hooper and Shoemaker, 1986; Kirchner, 2003). Over longer timescales, the damping and phase lagging of the seasonal isotopic cycle
directly imply that a fraction of each season's precipitation is stored in
the catchment (as snowpack, soil water, or deeper groundwater, for example),
eventually becoming streamflow in future seasons. But how much winter
precipitation eventually becomes summer streamflow (for example), and vice
versa? How much summer (or winter) precipitation eventually evapotranspires?
Quantitative answers to questions like these can shed light on how
catchments store and partition water on seasonal timescales.
Our goal is to quantify how precipitation is partitioned between streamflow
and evapotranspiration, both within an individual season and between
seasons. Figure 3 shows the seasonal cycles in precipitation and streamflow
isotopes at Watershed 3, averaged over the entire period of record. Monthly
average isotope signatures in precipitation (dark blue symbols in Fig. 3a)
reveal two isotopically distinct seasons: a 4-month snow-dominated winter
(December through March, with isotopically light precipitation) and an
8-month rain-dominated summer (April through November, with isotopically
heavy precipitation). We base our analysis on these two seasons, despite
their different lengths, because the results will be most precise if the two
inputs are as isotopically distinct as possible. These two seasons coincide
with monthly mean air temperatures above and below freezing (gray reference
line in Fig. 3f). Here we will refer to either the snowy and rainy seasons,
or winter and summer, interchangeably, but neither end-member mixing nor
end-member splitting requires the winter season to be snow-dominated.
Seasonal variation in deuterium ratios in bulk samples of
precipitation (a, c) and grab samples of streamflow
(b, d) from 2006 through 2010 at Hubbard Brook Watershed 3.
Diamonds in panel (e) are monthly water fluxes averaged over 1958–2014,
showing distinct effects of snowmelt in March through May, and
evapotranspiration in June through September. Diamonds in panel (f) are
monthly mean air temperatures relative to gray reference line of 0 ∘C. Light blue dots in panels (a–d) show individual samples, with
3 or 4 years of sampling overlapped, depending on month. Dark blue dots show
monthly volume-weighted means; error bars show standard errors where these
are larger than plotting symbols. Gray dashed line shows the volume-weighted
mean for all precipitation. Horizontal bars show seasonal volume-weighted
precipitation means ± standard errors, using two different definitions
of seasons. (a, b) show seasons defined by the break in
isotopic composition between months in which precipitation is predominantly
rain (April–November) and predominantly snow (December–March). Defining the
seasons in this way maximizes the isotopic difference between them. The next
two plots (c, d) show the same underlying isotope measurements, but with
averages defined for the growing season (June–September) and the dormant
season (October–May). These seasons are isotopically less distinct than the
rainy/snowy seasons, because the dormant season overlaps the isotopic shifts
between November-December and March–April. The seasonal precipitation means
are copied in the right-hand plots (along with the individual precipitation
values themselves, in gray), for comparison with the streamflow isotope
measurements. Streamflow separation into rainy-season vs. snowy-season
precipitation sources is more precise, because these seasonal precipitation
sources are more distinct, in comparison to growing-season vs.
dormant-season precipitation sources.
Seasonal origins of summer and winter streamflow
The damping of the seasonal precipitation isotopic cycle, as seen in Fig. 2a, implies that streamflow during each season must represent a mixture of
precipitation from both seasons, potentially spanning multiple years. We can
use conventional end-member mixing analysis to straightforwardly estimate
how summer and winter precipitation combine to form seasonal streamflow.
Because the two seasons are defined such that they span the entire year,
stream discharge in each season must be derived from a combination of summer
and/or winter precipitation:
Qs=qPs→Qs+qPw→Qs,Qw=qPs→Qw+qPw→Qw,
where Qs and Qw represent the average annual
sums of stream discharge during the summer and winter seasons, and (for
example) qPs→Qs and
qPw→Qs are the average
annual fluxes of summer streamflow that originated as summer and winter
precipitation, respectively. Equation (9) directly implies that, no matter
how the precipitation end-members are defined, they must jointly account for
all the precipitation that could eventually become streamflow (including,
potentially, precipitation in multiple previous summers or winters). In
other words, streamflow must be composed only of a mixture of the summer and
winter precipitation, Ps and Pw; there can be no
other end-members, sampled or not (although obviously streamflow can contain
flows from various catchment compartments in which summer and winter
precipitation have been stored and mixed). We also assume isotopic mass
balance for the water that eventually becomes discharge:
Qsδ‾Qs=qPs→Qsδ‾Ps+qPw→Qsδ‾PwandQwδ‾Qw=qPs→Qwδ‾Ps+qPw→Qwδ‾Pw,
where δ‾Qs, δ‾Qw, δ‾Ps, and
δ‾Pw are the volume-weighted average
isotopic signatures in summer and winter streamflow and precipitation.
Equation (10) implies that the precipitation that eventually becomes
streamflow does not undergo substantial isotopic fractionation (the effects
of which are discussed further in Sect. 3.3). It does not imply that no such
fractionation occurs in the water fluxes that are eventually evapotranspired
(and in any case, evapotranspiration fluxes are neither sampled nor directly
measured). Combining Eqs. (9) and (10) yields the end-member mixing
equations for summer streamflow,
fQs←Ps=qPs→QsQs=δ‾Qs-δ‾Pwδ‾Ps-δ‾PwandfQs←Pw=qPw→QsQs=δ‾Qs-δ‾Psδ‾Pw-δ‾Ps,
where fQs←Ps and
fQs←Pw represent
the fractions of summer streamflow that originated as summer and winter
precipitation, respectively. An analogous pair of end-member mixing
equations can be used to estimate the fractions of winter streamflow that
originate as summer and winter precipitation.
Partitioning of precipitation (P) into streamflow (Q) and
evapotranspiration (ET) during the snow-dominated season (December–March)
and the rain-dominated season (April–November), inferred from annual water
fluxes and volume-weighted δ2H at Hubbard Brook Watershed 3.
Essentially all evapotranspiration is derived from rainy-season
precipitation. Roughly half of rainy-season precipitation eventually
evapotranspires, about one-third eventually becomes rainy-season streamflow,
and about one-sixth eventually becomes snowy-season streamflow. Only about
one-fourth of snowy-season precipitation becomes snowy-season streamflow,
with about half becoming rainy-season streamflow and perhaps one-fifth being
lost to evaporation and transpiration. Roughly half of each season's
streamflow is derived from the other season's precipitation, implying
substantial inter-seasonal storage in snowpacks or groundwaters. All
quantities are shown ± standard errors. Widths of lines are
approximately proportional to water fluxes. Fluxes within 1 standard error
of zero are shown by dashed lines. Percentages may not add up to 100 due to
rounding.
As Fig. 4 shows, Eq. (11) and the isotope data from Watershed 3 imply that
about 38 % of summer (rainy-season) streamflow originates as winter
(snowy-season) precipitation, and 62 % originates as rainy-season
precipitation. They also imply that about 40 % of winter (snowy-season)
streamflow originates as snowy-season precipitation and 60 % as
rainy-season precipitation. These percentages should be assessed in
comparison with the proportions of precipitation that originate in the snowy
and rainy seasons. At Watershed 3, the rainy season comprises two-thirds of
the year and 70 % of total precipitation, as a long-term average. If
summer and winter streamflow were derived proportionally from each season's
precipitation, each would consist of 70 % rainy-season precipitation and
30 % snowy-season precipitation. Using these percentages as a reference
point, we can quantify how the contributions of summer and winter
precipitation to streamflow deviate from their shares of total
precipitation, using relationships of the form
ΔfQs←Ps=fQs←Ps-PsPPsP=δ‾Qs-δ‾Pwδ‾Ps-δ‾PwPPs-1andΔfQs←Pw=fQs←Pw-PwPPwP=-ΔfQs←PsPsPw,
where ΔfQs←Ps
and ΔfQs←Pw
are the fractional over- or under-representation of each season's
precipitation in summer streamflow. These calculations yield the result that
winter precipitation is over-represented by 26 % and 32 % (and summer
precipitation is under-represented by 11 % and 14 %) in summer and
winter streamflow, respectively. The under-representation of summer
precipitation in both seasons' streamflow implies that it is
over-represented in evapotranspiration (as examined in Sect. 2.3 below).
More generally, the isotope data from Watershed 3 imply that substantial
fractions of streamflow are derived from water that has been stored in the
catchment from previous seasons as either snowpack or groundwater (and, in
the case of groundwater, potentially also including water from previous
years). Many hydrograph separation studies, including the work of Hooper and
Shoemaker (1986) at Watershed 3, have shown that streamflow is often
composed primarily of pre-event water. The results in this section, which
can be loosely considered to be a seasonal-scale hydrograph separation, extend
the previous event-scale findings by showing that even at the seasonal timescale, streamflow is not clearly dominated by current (i.e., same-season)
precipitation.
Seasonal origins of evapotranspiration
We can straightforwardly extend the seasonal end-member mixing approach
above to estimate how much evapotranspiration originates as summer vs.
winter precipitation. We begin by assuming that the water fluxes satisfy
mass balance:
Ps+Pw=Q+ET,
where Ps and Pw represent the average annual
sums of precipitation falling in the summer and winter, respectively, Q
represents annual average discharge, and ET represents average annual
evapotranspiration. Equation (13) assumes that these fluxes are much larger
than any other inputs (such as direct surface condensation or groundwater
inflows) or outputs (such as groundwater outflow). Equation (13) is also
assumed to hold over timescales long enough that changes in catchment
storage are trivial compared to the cumulative input and output fluxes.
These same assumptions are invoked in hydrometric studies that infer ET
from long-term catchment water balances (e.g., Vadeboncoeur et al., 2018).
However, such hydrometric studies cannot reliably estimate the seasonal
origins of evapotranspiration, because changes in catchment storage may be
substantial on seasonal timescales.
We can straightforwardly apply end-member mixing to the total annual
discharge, analogously to the approach used in Eqs. (9)–(11) for discharge
during the individual seasons. All discharge must originate as either summer
or winter precipitation, and thus
Q=qPs→Q+qPw→Q,
where qPs→Q and
qPw→Q are the annual average fluxes
that originate as summer and winter precipitation. Isotopic mass balance for
the water that eventually becomes discharge implies
Qδ‾Q=qPs→Qδ‾Ps+qPw→Qδ‾Pw,
where δ‾Q is the volume-weighted isotopic signature
of total annual streamflow. Jointly solving Eqs. (14) and (15) yields the
seasonal end-member mixing equations for total annual streamflow,
fQ←Ps=qPs→QQ=δ‾Q-δ‾Pwδ‾Ps-δ‾PwandfQ←Pw=qPw→QQ=δ‾Q-δ‾Psδ‾Pw-δ‾Ps
where fQ←Ps and
fQ←Pw represent the fractions
of total annual streamflow that originate as summer and winter
precipitation, respectively. Using the input data shown in Fig. 4, Eq. (16)
yields the result that average annual streamflow is composed of 57±7 % rainy-season precipitation and 43±7 % snowy-season
precipitation.
What does this have to do with evapotranspiration? A consequence of the
assumed water balance closure (Eq. 13) is that all precipitation must
eventually become either evapotranspiration or discharge, that is,
Ps=qPs→Q+qPs→ET,Pw=qPw→Q+qPw→ET,
where qPs→Q and
qPs→ET (for example) represent the
average annual fluxes of discharge and streamflow that originate as summer
precipitation (potentially including summer precipitation in previous
years). Thus summer and winter precipitation that does not eventually become
streamflow must contribute to evapotranspiration. Combining Eqs. (13), (16),
and (17), one directly obtains the fraction of ET originating as summer
precipitation, fET←Ps:
fET←Ps=qPs→ETET=Ps-qPs→QPs+Pw-Q=Ps-QfQ←PsPs+Pw-Q=Ps-Qδ‾Q-δ‾Pwδ‾Ps-δ‾PwPs+Pw-Q.
An analogous expression can be used to estimate fET←Pw, the fraction of ET originating as winter
precipitation.
As Fig. 4 shows, Eq. (18) implies that evapotranspiration at Watershed 3 is
almost entirely (85±15 %) derived from rainy-season precipitation,
and the fraction derived from snowy-season precipitation is not
distinguishable from zero (15±15 %). This result is not
particularly surprising, for several reasons. First, the rainy season is
twice as long as the snowy season, and accounts for 70 % of total annual
precipitation. Second, the higher temperatures and vapor pressure deficits
that prevail during the summer imply that both surface evaporation rates and
potential evapotranspiration rates will be higher during the rainy season.
Third, the growing season of Watershed 3's mixed hardwood forest occurs
during the rainy season, implying that transpiration rates during the snowy
season should be small. Thus the results of Eq. (18) are biologically and
climatologically plausible.
It should be noted that although the lopsided ET source attribution shown in
Fig. 4 is not surprising, neither is it intuitively obvious. Intuitively one
might assume that since streamflow at Watershed 3 is a mixture of roughly
equal fractions of summer and winter precipitation, they should also each
comprise roughly half of evapotranspiration. The isotopic mass-balance
calculation in Eq. (18) shows that this intuition is wrong, and it also
suggests why: annual ET is considerably smaller than annual Q, and winter
precipitation is considerably smaller than summer precipitation (partly
because the summer is twice as long). Thus winter precipitation can be
greatly under-represented in ET while also being roughly half (in fact, less
than half) of discharge.
Following the approach in Eq. (12), we can quantify the fractional over- or
under-representation of summer and winter precipitation in total (summer
plus winter) streamflow as
ΔfQ←Ps=fQ←Ps-PsPPsP=δ‾Q-δ‾Pwδ‾Ps-δ‾PwPPs-1andΔfQ←Pw=fQ←Pw-PwPPwP=-ΔfQ←PsPsPw
and the fractional over- or under-representation of summer and winter
precipitation in total ET as
ΔfET←Ps=fET←Ps-PsPPsP=Ps-Qδ‾Q-δ‾Pwδ‾Ps-δ‾PwPs+Pw-QPPs-1andΔfET←Pw=fET←Pw-PwPPwP=-ΔfET←PsPsPw.
These calculations yield the result that summer precipitation is
under-represented by 19 % in annual streamflow (summer precipitation is
70 % of annual precipitation but only 61 % of annual streamflow, so
summer precipitation is under-represented in streamflow by 19 %), and
winter precipitation is over-represented by 28 %. By contrast, winter
precipitation is under-represented in ET by 50 % (winter precipitation
accounts for 30 % of annual precipitation but only 15 % of ET, or only
about half of ET's share of total precipitation), and summer precipitation
is over-represented by 22 %.
Finally, it is worth noting that one can infer the average isotopic
composition of the unmeasured ET flux straightforwardly by isotope mass
balance,
δ‾ET=Psδ‾Ps+Pwδ‾Pw-Qδ‾QPs+Pw-Q.
If the associated uncertainties are acceptably small (see error propagation
in the Supplement), inferred values of δ‾ET could
be useful in interpreting tree-ring isotopic records. Tree-ring isotope
values are often assumed to reflect the isotopic composition of either
growing-season precipitation or annual average precipitation, but the
seasonal sources of xylem water (and thus of tree-ring isotopes) may vary
with climate and subsurface moisture storage characteristics. Thus, if
δ‾ET reflects the isotopic composition of the
transpiration flux (and thus of xylem water), it would provide an additional
constraint for calibrating tree-ring isotopes. Inferred values of
δ‾ET could also be useful in quantifying the
relative contributions of evaporation and transpiration to ET at
whole-catchment scale, if one can also directly measure the isotopic
composition of the evaporation and transpiration fluxes (through soil and
xylem sampling, for example).
End-member splitting of seasonal precipitation into seasonal discharge
and evapotranspiration
Up to this point we have analyzed evapotranspiration and seasonal discharge
as mixtures of summer and winter precipitation. In this section, we analyze
the corresponding question of how summer and winter precipitation is
partitioned among these outputs. That is, having addressed the question of
where the outputs come from, we now address the mirror-image question of
where the inputs go. Mathematically this can be accomplished by re-scaling
the end-member mixing results by the ratios of output fluxes to input
fluxes, as introduced in Sect. 1. Consider, for example, the annual
average flux qPs→Qs of
summer precipitation that becomes summer streamflow. This flux, divided by
the annual sum of summer streamflow (the total output flux), yields
fQs←Ps, the
fraction of summer streamflow that originated as summer precipitation (Eq. 11). But this same flux, when divided by the annual sum of summer precipitation
(the total input flux), yields the fraction of summer precipitation that
eventually becomes summer streamflow. This fraction, here denoted ηPs→Qs, can
therefore be directly calculated from fQs←Ps by multiplying by the ratio of the output flux to
the input flux:
ηPs→Qs=qPs→QsPs=QsPsqPs→QsQs=QsPsfQs←Ps=QsPsδ‾Qs-δ‾Pwδ‾Ps-δ‾Pw.
Similar relationships can be used to calculate the fraction of summer
precipitation that eventually becomes winter streamflow,
ηPs→Qw=qPs→QwPs=QwPsqPs→QwQw=QwPsfQw←Ps=QwPsδ‾Qw-δ‾Pwδ‾Ps-δ‾Pw,
the fraction that eventually becomes streamflow in either season,
ηPs→Q=qPs→QPs=QPsqPs→QQw=QPsfQ←Ps=QPsδ‾Q-δ‾Pwδ‾Ps-δ‾Pw,
and the fraction that is eventually evapotranspired,
ηPs→ET=qPs→ETPs=ETPsfET←Ps=1-ηPs→Q=1-QPsfQ←Ps=1-QPsδ‾Q-δ‾Pwδ‾Ps-δ‾Pw.
Analogous equations can be used to partition winter precipitation among the
same outputs. Intriguingly, Eq. (25) does not require calculation of the mass
balance ET=Ps+Pw-Q; thus one can calculate the
fraction of each season's precipitation that is eventually transpired, even
if the evapotranspiration rate itself is not well constrained by mass
balance.
As Fig. 4 shows, Eqs. (22)–(25) imply that roughly half (44±8 %) of
rainy-season precipitation is eventually evapotranspired. The remainder is
partitioned between summer and winter streamflow in roughly a 2:1 ratio
(39±6 % and 18±3 % of rainy-season precipitation,
respectively). By contrast, much less (and perhaps none at all) of
snowy-season precipitation (18±18 %) is eventually evapotranspired,
although the remainder is split between summer and winter streamflow in
nearly the same 2:1 ratio (55±13 % and 27±6 %,
respectively) as the rainy-season precipitation is partitioned. This 2:1
ratio is perhaps unsurprising, because the summer season is twice as long as
the winter season, and summer streamflow is 68 % of total streamflow, but
it implies significant carryover of water from each season to the next.
Figure 4 illustrates how end-member mixing and end-member splitting yield
different (but complementary) perspectives on the catchment water balance.
Only about half of rainy-season precipitation is eventually evapotranspired,
but nearly all evapotranspiration originates as rainy-season precipitation.
The two proportions are different but not inconsistent, for the simple
reason that rainy-season precipitation is much greater than annual
evapotranspiration. Likewise, both rainy-season and snowy-season
precipitation are split between rainy- and snowy-season streamflow in a 2:1
ratio, but streamflow during both seasons originates from roughly equal
proportions of snowy- and rainy-season precipitation. Again the proportions
are different but not inconsistent, since total rainfall and total
streamflow are both greater during the rainy season than during the snowy
season.
As with the mixing fractions derived in Sects. 2.2 and 2.3, we can also
express end-member splitting proportions in terms of how much the possible
fates of precipitation are over- or under-represented, relative to their
flow-proportional share of total precipitation. For example, from Fig. 4 one
can see that roughly one-third of summer precipitation ultimately becomes
summer streamflow; is this more, or less, than one would expect if
precipitation were split among all of its fates proportionally to their
total fluxes? If precipitation were split proportionally among summer
streamflow, winter streamflow, and evapotranspiration, and if summer and
winter precipitation were both split by the same proportions, then the
proportion of precipitation that ultimately became summer streamflow would
be QsP=0.44. This provides a reference point for
comparing the actual end-member splitting result of ηPs→Qs=39±6 %:
ΔηPs→Qs=ηPs→Qs-QsPQsP=PPsfQs←Ps-1=PPsδ‾Qs-δ‾Pwδ‾Ps-δ‾Pw-1=ΔfQs←Ps.
It may seem strange that ΔηPs→Qs, the fractional over- or under-representation of
summer streamflow as a fate for summer precipitation, is numerically equal
to ΔfQs←Ps, the fractional over- or under-representation of
summer precipitation in summer streamflow. This is particularly so, given
that the end-member splitting proportion ηPs→Qs (Eq. 22) is
substantially different from the end-member mixing fraction
fQs←Ps
(Eq. 11), and the two metrics are compared to two different reference points
(QsP for ηPs→Qs and PsP for
fQs←Ps).
However, because the ratio between these reference points is
QsPs and the ratio between ηPs→Qs and
fQs←Ps is
also QsPs, it follows mathematically that
ΔηPs→Qs=ΔfQs←Ps.
The same phenomenon holds for the under- or over-representation of winter
streamflow as a fate of summer precipitation, for which an appropriate point
of reference is QwP,
ΔηPs→Qw=ηPs→Qw-QwPQwP=PPsfQw←Ps-1=PPsδ‾Qw-δ‾Pwδ‾Ps-δ‾Pw-1=ΔfQw←Ps,
and the under- or over-representation of annual streamflow as a fate of
summer precipitation, for which an appropriate point of reference is
QP,
ΔηPs→Q=ηPs→Q-QPQP=PPsfQ←Ps-1=PPsδ‾Q-δ‾Pwδ‾Ps-δ‾Pw-1=ΔfQ←Ps,
and the under- or over-representation of evapotranspiration as a fate of
summer precipitation, for which an appropriate point of reference is
ETP:
ΔηPs→ET=ηPs→ET-ETPETP=PPsfET←Ps-1=PPsPs-Qδ‾Q-δ‾Pwδ‾Ps-δ‾PwPs+Pw-Q-1=ΔfET←Ps.
Naturally, one can also write analogous expressions for the corresponding
fractions of winter precipitation. Using Eqs. (26)–(29) and the information
in Fig. 4, one can calculate that the fractions of summer precipitation
going to summer and winter streamflow are 11 % and 14 % less, and the
fraction going to ET is 22 % greater, than their proportional shares of
total precipitation. By contrast, the fractions of winter precipitation
going to summer and winter streamflow are 26 % and 31 % greater, and the
fraction going to ET is 50 % less, than their proportional shares of total
precipitation. These percentages do not balance because they are percentages
of different quantities (the proportions of total outflows).
Stepping back from these details, however, the most striking result of the
end-member splitting analysis is that 18 % of rainy-season precipitation
(or 160 mm yr-1), and 55 % of snowy-season precipitation (or 219 mm yr-1), leaves the catchment as streamflow during a different
season than the one that it fell in. This reinforces the point that there
must be significant inter-seasonal water storage at the catchment scale. The
annual snowpack clearly represents a significant inter-seasonal storage of
winter precipitation, because much of its melt takes place in April, which
is during the rainy season. Annual peak snowpack storage is roughly 190 mm
of snow water equivalent (Campbell et al., 2010), which equals roughly half
of average winter precipitation, and apparently a substantial fraction of
this crosses into the rainy season to become streamflow (for example, during
the snowmelt pulse in April), but only a small fraction is evapotranspired.
End-member splitting calculations are based on mass balances, and therefore
must be applied to long-term average fluxes, for which mass balances can be
assumed to be reasonably precise. The calculations outlined in this section
further assume that the sampled precipitation and streamflow are
representative of the snowy and rainy seasons. Of course, the inputs to any
such calculation will inevitably be based on finite sets of samples and
measurements, which may deviate somewhat from the (unknown) long-term
averages. How sensitive are the results to the specific periods that we
analyzed? How much uncertainty would be introduced if the available records
were even more limited? To get some idea, we extracted three individual
water years, each running from December to November (and thus each including
one snowy season and one rainy season), from the isotope and water flux time
series. We then repeated the end-member splitting analysis using only data
from each individual water year (daily precipitation and discharge fluxes,
and a total of roughly 24 biweekly isotope measurements in precipitation and
streamflow). The results are shown in Fig. 5, which also compares end-member
splitting proportions obtained from oxygen-18 (shown by circles) with those
obtained from deuterium (shown by diamonds). Figure 5 shows that when one
uses shorter data sets (light blue symbols) the resulting uncertainties are
bigger, as expected, but the error bars overlap with the estimates derived
from the entire data set (dark blue symbols, based on all available isotope
data, and long-term average water fluxes). These results demonstrate that
the small-sample estimates are realistic approximations (within their
standard errors) of the values that would be derived from the more complete
data set.
Seasonal partitioning of precipitation (P) into streamflow (Q) and
evapotranspiration (ET), estimated from δ18O (circles) and
δ2H (diamonds) from individual water years. Solid symbols show
results using all available isotope measurements and long-term averages of P and Q water fluxes. Open symbols show results using only isotope and water flux measurements collected during individual water years (2007 through
2009, from left to right). Water years are defined from December through the
following November, thus including one snowy season and the following rainy
season. Seasonal partitioning estimates derived from δ18O and
δ2H generally agree within their standard errors, as do
estimates derived from individual years of data (open symbols).
Unsurprisingly, estimates derived from individual years have larger
uncertainties than those derived from all available data.
Partitioning of seasonal precipitation into monthly discharges
Because we have only one tracer in practice (we nominally have both
oxygen-18 and deuterium, but they are largely redundant with one another),
end-member mixing can quantify the fractional contributions from only two
sources (such as summer and winter precipitation) in each mixture (such as
summer and winter streamflow). There is, however, no mathematical limit to
the number of different mixtures that such end-member mixing calculations
could be applied to. (There may be a logical limit, of course; it would make
little sense to express streamflow on each individual day as a mixture of
summer and winter precipitation, given the wide variability in precipitation
isotopes from one storm to the next.) Because there is no mathematical limit
on the number of different mixtures, in the context of end-member splitting
there is no mathematical limit on the number of different fates that each
source can be partitioned among. The only constraint is that the outputs
must jointly account for all of the input (i.e., all of the precipitation
must go somewhere), and we must have tracer and water flux measurements for
all-but-one of them. In most practical cases, the unmeasured output will be
evapotranspiration (or will be called
evapotranspiration, although it will formally be the sum of all unmeasured
fluxes).
Here we illustrate this approach by splitting summer and winter
precipitation among each month's streamflow, instead of just summer and
winter streamflow. The monthly end-member mixing equations are of the form
fQi←Ps=qPs→QiQi=δ‾Qi-δ‾Pwδ‾Ps-δ‾PwandfQi←Pw=δ‾Qi-δ‾Psδ‾Pw-δ‾Ps,
where Qi is the monthly discharge in month i. The corresponding
end-member splitting equations, derived by the logic of Eq. (4), are
ηPs→Qi=qPs→QiPs=QiPsfQi←Ps=QiPsδ‾Qi-δ‾Pwδ‾Ps-δ‾PwandηPw→Qi=QiPwδ‾Qi-δ‾Psδ‾Pw-δ‾Ps.
The results of this analysis are shown in Fig. 6. Although monthly
precipitation rates are roughly equal throughout the year, monthly discharge
rates show a distinct snowmelt-driven peak in April and distinct low flows
attributable to evapotranspiration in July, August, and September (Fig. 6a).
Monthly end-member mixing (Eq. 30) shows that the mixing fraction
fQi←Ps of summer
precipitation in streamflow reaches a minimum of 34 % during the spring
discharge peak and increases throughout the growing season, peaking at
88 % in August (Fig. 6b). The partitioning ηPs→Qi of summer
precipitation among monthly streamflows, however, shows a very different
pattern, peaking during spring snowmelt (when the fraction of summer
precipitation in streamflow is lowest) and reaching a minimum during the
growing season (when the fraction of summer precipitation in streamflow is
highest; Fig. 6c).
Patterns in monthly average precipitation and streamflow fluxes (a), isotope hydrograph separations of rainy- and snowy-season precipitation
in monthly streamflows (b), and distributions of rainy-season (c) and
snowy-season (d) precipitation in streamflow (fraction of precipitation
leaving as streamflow in each month). Proportions in (c) and (d) do not sum
to 100 % because they do not include evapotranspiration losses (which are
18 % and 44 % of snowy-season and rainy-season precipitation,
respectively). Average precipitation fluxes vary little from month to month,
whereas average streamflow fluxes show clear high flows resulting from
snowmelt from March through May and clear low flows attributable to
evapotranspiration losses from July through September (a). Both intervals
are marked by gray shading. Monthly isotope hydrograph separations (b) show
larger fractions of snowy-season precipitation in streamflow during the
snowmelt period, followed by a steadily growing fraction of rainy-season
precipitation that reaches a peak of nearly 90 % in August. However, much
more rainy-season precipitation becomes streamflow during snowmelt (c), when
its fractional contribution to streamflow is lowest (b), than during late
summer, when its fractional contribution to streamflow is relatively high (b, c). This occurs because monthly total streamflow is much higher during
snowmelt than during the high-ET conditions of late summer. A relatively
large proportion of rainy-season precipitation also becomes streamflow in
October through December, as monthly total streamflow recovers after the end
of the summer ET peak. The proportion of snowy-season precipitation becoming
streamflow (d) unsurprisingly peaks in during peak snowmelt, when monthly
streamflow is highest and the fractional contribution of snowy-season
precipitation to that streamflow is likewise high.
This relationship arises because, as Eq. (31) shows, the “forward”
partitioning fractions ηPs→Qi of precipitation (Fig. 6c) are proportional to the
“backward” mixing fractions fQi←Ps (Fig. 6b), which vary by less than a factor of
3, multiplied by the monthly discharges Qi (Fig. 6a), which vary by
nearly a factor of 9. Because Qi is more variable than
fQi←Ps, variations in the
“forward” partitioning fractions ηPs→Qi largely reflect variations in Qi. For example,
between April and August the percentage of rainy-season precipitation in
streamflow increases from 34 % to 88 % (a factor of 2.5), but the total
discharge flux decreases from 205 to 26 mm month-1 (a factor of
nearly 8). Thus although rainy-season precipitation makes up a greater
fraction of streamflow in August than in April, August streamflow accounts
for a much smaller fraction of rainy-season precipitation than April
streamflow does. The same principle also holds for the “forward”
partitioning fractions ηPw→Qi of winter precipitation, but in this case it is less
evident because the seasonal patterns in Qi and the “backward” mixing
fractions fQi←Pw of winter
precipitation generally reinforce, rather than offset, one another.
Unsurprisingly, the forward partitioning fractions ηPw→Qi of winter
precipitation among monthly discharges reach their peak during spring
snowmelt and their minimum during summer low flows.
The forward partitioning fractions ηPs→Qi of summer
precipitation reach a second peak in late autumn, after the end of the
growing season but before substantial snowfall (Fig. 6c). During this
period, interception and transpiration losses are relatively small, as one
can see from the rise in stream discharge from September through November
despite nearly constant monthly precipitation totals (Fig. 6a). Thus late
autumn streamflows are relatively high. Because those streamflows also
contain large mixing fractions fQi←Ps of summer precipitation (Fig. 6b), they result in
a peak in the end-member splits of summer precipitation ηPs→Qi (Fig. 6c). Somewhat
surprisingly, the partitioning fractions ηPw→Qi of winter
precipitation also rise somewhat in late autumn, even though the winter
season ended more than six months ago (Fig. 6d), and precipitation does not
acquire its winter isotopic signature again until December. This rise in the
late autumn occurs because snowy-season precipitation still makes up roughly
15 % of streamflow (Fig. 6b), presumably reflecting long-term subsurface
storage mobilized by increased infiltration of autumn rainfall after the
growing season ends.
In any case, the most striking feature of Fig. 6 is that it indicates that
substantial export of rainy-season precipitation occurs just as the snowy
season is ending and the rainy season is beginning. This could result from
the big April snowmelt pulse mobilizing groundwater that was stored through
the winter. Alternatively, it could result from the snowmelt pulse
saturating shallow soil layers and causing large fractions of April rainfall
to reach the stream. The fraction of summer precipitation in April
streamflow is 34±11 %, or 69±23 mm month-1 out of an
average April streamflow of 205±5 mm month-1. This 69±23 mm month-1 must consist of April precipitation, or precipitation
from previous summers, or a mixture of both. If the 69±23 mm month-1 were composed entirely of April precipitation, it would
account for about 70 % of average April precipitation (106±5 mm month-1). Thus these results do not require that large quantities of
summer precipitation must have overwintered as groundwater, but they also do
not exclude that possibility.
End-member splitting of growing-season and dormant-season precipitation
In the analysis presented above in Sect. 2.2–2.5, we separated the year
into a rainy season and a snowy season, to maximize the isotopic difference
between the two precipitation end-members. Other precipitation seasons,
which are less optimal from an isotopic separation standpoint, are also
possible. It could be of biological interest, for example, to separate the
year into the growing season (June–September) and the dormant season
(October–May). The analysis proceeds exactly as described in Eqs. (9)–(29),
except now “summer” and “winter” correspond to the growing and dormant
seasons, respectively. As Fig. 3c–d show, the precipitation isotopes in
the growing and dormant seasons are less distinct than those in the rainy
and snowy seasons, for the simple reason that the dormant season includes
both rain-dominated months (October–November and April–May) and
snow-dominated months (December–March). As a consequence, mixing fractions
and end-member splits calculated from the growing-season and dormant-season
end-members will inevitably have larger uncertainties than those calculated
from the rainy- and snowy-season end-members. Nonetheless, as Fig. 7 shows,
one can still draw useful inferences from such end-member mixing and
splitting calculations. From Fig. 7 one can see that nearly all (84±21 %) of dormant-season streamflow originates from dormant-season
precipitation, and the contribution from growing-season precipitation is
zero within error (16±21 %). Conversely, roughly half (45±19 %) of growing-season streamflow originates from dormant-season
precipitation, and the other half (55±19 %) originates from
growing-season precipitation. Evapotranspiration appears to be mostly
(60±35 %) derived from growing-season precipitation, with a smaller
contribution (40±35 %) from dormant-season precipitation, but the
uncertainties are large enough that many other mixing fractions are also
possible. End-member splitting shows that a large fraction (72±18 %) of dormant-season precipitation eventually becomes dormant-season
streamflow, with a small but well-defined fraction (6±2 %)
eventually becoming growing-season streamflow and a larger but uncertain
fraction (22±19 %) potentially being evapotranspired. Conversely, a
large but uncertain fraction (62±36 %) of growing-season
precipitation is eventually evapotranspired, with a small but well-defined
fraction (14±5 %) eventually becoming growing-season streamflow and
a small and highly uncertain fraction (24±32 %) becoming
dormant-season streamflow.
Partitioning of precipitation (P) into streamflow (Q) and
evapotranspiration (ET) during the dormant season (October–May) and the
growing season (June–September), inferred from annual water fluxes and
volume-weighted δ2H at Hubbard Brook Watershed 3. These two
precipitation seasons are less isotopically distinct than the rainy/snowy
seasons (see Fig. 3), so the propagated uncertainties are correspondingly
larger than those shown in Fig. 4. Evapotranspiration is mostly derived from
growing-season precipitation, with a smaller fraction coming from
dormant-season precipitation, but both percentages are highly uncertain.
Most growing-season precipitation is eventually evapotranspired, with a
small but well-defined fraction eventually becoming growing-season
streamflow. Roughly half of growing-season streamflow is derived from a
small but well-defined fraction of dormant-season precipitation. Most of the
rest of dormant-season precipitation eventually becomes dormant-season
streamflow, and about one-fifth may evapotranspire (although this is highly
uncertain). All quantities are shown ± standard errors. Widths of
lines are approximately proportional to water fluxes. Fluxes within 1
standard error of zero are shown by dashed lines.
It is noteworthy that, in Fig. 7, dormant-season precipitation makes up
about half (45±19 %) of growing-season discharge, and nearly all
(79±20 %) of total annual discharge, but probably less than half
(40±35 %) of evapotranspiration. Conversely, growing-season
precipitation probably makes up the bulk (60±35 %) of
evapotranspiration, but only a small fraction (21±20 %) of total
annual discharge. This example illustrates how an isotopic separation
between “blue water” and “green water” (the so-called “two water worlds”
phenomenon) could arise through unsurprising contrasts between the
proportions of winter and summer precipitation that eventually become
evapotranspiration vs. streamflow. We emphasize that this analysis makes no
specific inference about how, mechanistically, such a separation occurs.
Importantly, however, this isotopic separation does not require that “blue
water” and “green water” are sourced from physically distinct storages. In
particular, it does not require a separation between “bound waters” that
primarily supply ET and “mobile waters” that primarily supply streamflow
(Brooks et al., 2010; Good et al., 2015), although it also does not rule
this out. Instead, our analysis shows that isotopic evidence of apparent
“two water worlds” requires only that evapotranspiration rates vary
seasonally, and that catchments do not store enough water to average out the
isotopic differences between summer and winter precipitation when those
waters become ET or streamflow. These conditions are likely to be met in
many catchments.
As a further thought experiment, we can ask how snowy- and rainy-season
precipitation contribute to – and are partitioned among – dormant- and
growing-season streamflow. Here we make use of the fact that the analyses
derived above do not require us to use the same seasons to characterize
precipitation and streamflow. Thus we can repeat the same analysis that is
outlined in Eqs. (9)–(29), using “summer” to refer to growing-season
(June–September) streamflow but rainy-season (April–November) precipitation,
and “winter” to refer to dormant-season (October–May) streamflow but
snowy-season (December–March) precipitation. Naturally, one must keep in
mind the different lengths of these seasons, as well as their sometimes
substantial differences in water fluxes, when interpreting the results.
The results of this analysis are shown in Fig. 8. Just as in Fig. 4,
evapotranspiration is derived almost entirely (85±15 %) from
rainy-season precipitation, and relatively little, or almost not at all
(15±15 %), from snowy-season precipitation. These results are
identical to those obtained in Sect. 2.3 because, in our analysis, ET is not
(and cannot be) differentiated by season (unless we have measurements of the
ET fluxes themselves or of their isotopic signatures). Thus we can
distinguish the seasonal origins of ET fluxes, but not the seasons in which
those ET fluxes occur. Figure 8 shows that growing-season streamflow is
derived in roughly a 4:1 ratio from rainy-season and snowy-season
precipitation (79±8 % and 21±8 %, respectively), whereas
dormant-season streamflow is derived from nearly equal contributions from
the two seasons (58±9 % and 42±9 %, respectively). Roughly
half of rainy-season precipitation eventually evapotranspires; a roughly
equal amount (46±7 %) becomes dormant-season streamflow, and a
small but well-constrained fraction (10±1 %) becomes growing-season
streamflow. It is striking that this 10 % fraction of rainy-season
precipitation makes up the dominant fraction (79±8 %) of
growing-season streamflow, but this simply reflects the fact that
rainy-season precipitation is nearly 8 times larger than growing-season
streamflow. This is partly due to substantial evapotranspiration losses
during the growing season and also due to the fact that the growing season
is only half as long as the rainy season. It may seem striking that about
4 times as much rainy-season precipitation becomes dormant-season
streamflow as becomes growing-season streamflow. However, this is not as
surprising as it first might seem, given that half of the rainy season
overlaps with the dormant season (April–May and October–November) and that
the other half of the rainy season (i.e., the growing season) is marked by
substantial evapotranspiration losses and very low streamflows. The great
majority (77±16 %) of snowy-season precipitation becomes
dormant-season streamflow, which is unsurprising because both the snowy
season and the snowmelt period are contained within the dormant season.
Thus, not only is evapotranspiration almost entirely sourced from
rainy-season precipitation over the three summers for which measurements are
available, it also appears that relatively little snowy-season precipitation
could compensate for ecosystem water shortages during summer droughts,
because most snowy-season precipitation becomes streamflow in the dormant
season. A small but well-defined fraction (6±2 %) of snowy-season
precipitation becomes growing-season streamflow, and a small and indefinite
fraction (17±18 %) evapotranspires. It is noteworthy that about
one-fifth of growing-season streamflow is derived from snowy-season
precipitation, despite the fact that the growing season begins 2 months after
the snowy season ends. Thus this fraction of snowy-season precipitation
(roughly 25 mm yr-1) must be stored in the subsurface for at least
several months before becoming growing-season streamflow.
Partitioning of snowy-season (December–March) and rainy-season
(April–October) precipitation (P) into evapotranspiration (ET) and
streamflow (Q) during the dormant season (October–May) and the growing
season (June–September), inferred from annual water fluxes and
volume-weighted δ2H at Hubbard Brook Watershed 3. About half of
rainy-season precipitation eventually evapotranspires, and this accounts for
almost all the annual evapotranspiration flux; the contribution from
snowy-season precipitation is zero within error. About 10 % of
rainy-season precipitation accounts for four-fifths of growing-season
streamflow, and the remaining (46 %) rainy-season precipitation accounts
for about half of dormant-season streamflow. About three-fourths of
snowy-season precipitation become dormant season streamflow, and perhaps
one-sixth eventually evapotranspires (but this is zero within error). A
small but well-defined proportion is also carried over to the growing
season, accounting for one-fifth of growing-season streamflow. All
quantities are shown ± standard errors. Widths of lines are
approximately proportional to water fluxes. Fluxes within 1 standard error
of zero are shown by dashed lines.
Comparison with sine-wave fitting and young water fractions
Sections 2.2–2.4 and 2.6 draw inferences concerning intra- and
inter-seasonal storage and transport by comparing seasonal isotopic
variations in precipitation and streamflow. Seasonal isotope cycles have
been used to infer timescales of catchment storage for more than 2 decades,
since at least the work of DeWalle et al. (1997). The damping of seasonal
isotopic cycles has recently been shown to quantify the average fraction of
streamflow that is younger than approximately 2–3 months, even in spatially
heterogeneous and nonstationary catchments (Kirchner, 2016a, b). This “young
water fraction” can provide a consistency check on the end-member mixing
results reported here, because the two methods involve different
calculations based on different assumptions, although they both use the same
data.
Deuterium time series in biweekly bulk samples of precipitation
(light blue) and grab samples of streamwater (dark blue), with superimposed
seasonal sinusoidal cycles fitted by volume-weighted least squares. The
vertical axis has been expanded to better show the seasonal cycles, with the
result that several precipitation values are not shown. The amplitudes of
the fitted seasonal cycles are
AP=194±34‰ and
AS=87±09‰ in precipitation and streamflow, respectively,
implying that the flow-weighted young water fraction (the fraction of
discharge that is younger than approximately 2–3 months) is
Fyw*=AS/AP=045±009. Rescaling Fyw* by the
ratio between the average annual discharge and precipitation fluxes yields
the flow-weighted young water fraction of precipitation (the fraction of
precipitation that is discharged in less than approximately 2–3 months),
PFyw*=Fyw*Q‾/P‾=0.29±0.06.
Figure 9 shows volume-weighted seasonal sinusoidal cycles fitted to the
deuterium time series in precipitation and streamflow. The ratio between the
volume-weighted seasonal cycle amplitudes in streamflow and precipitation
(AQ* and AP*, respectively) yields
the volume-weighted young water fraction Fyw*=AQ*/AP*, the proportion (by volume)
of streamflow that is younger than roughly 2–3 months. (Here we follow von
Freyberg et al. (2018a) in using an asterisk to denote volume-weighted
quantities.) The cycles in Fig. 9 imply a volume-weighted young water
fraction Fyw* of 45±9 %, which is broadly
comparable to the fQw←Pw=40±9 % of snowy-season Q that
originates as snowy-season P and the fQs←Ps=55±19 % of growing-season Q that
originates as growing-season P (both 4-month seasons), and also consistent
with the fQs←Ps=62±9 % of rainy-season Q that originates as rainy-season P and the
fQw←Pw=84±21 % of dormant-season Q that originates as dormant-season P (both 8-month
seasons). All of these different measures are of the same general magnitude,
although as one would expect, the longer seasons are associated with larger
fractions of same-season precipitation in streamflow.
Following the approach of Eq. (4), we can multiply the volume-weighted young
water fraction by the ratio between the average streamflow and average
precipitation to obtain the young water fraction of precipitation
PFyw*=Fyw*Q‾/P‾, the average fraction (by volume) of
precipitation that leaves the catchment as streamflow within 2–3 months. The
cycles in Fig. 9 imply that the young water fraction of precipitation
PFyw* is 0.29±0.06, which can
be compared to the ηPw→Qw=27±6 % of snowy-season precipitation
that becomes snowy-season streamflow and the ηPs→Qs=14±5 % of
growing-season precipitation that becomes growing-season streamflow (both
4-month seasons), or the ηPs→Qs=39±6 % of rainy-season precipitation
that becomes rainy-season streamflow and the ηPw→Qw=72±18 %
of dormant-season precipitation that becomes dormant-season streamflow (both
8-month seasons). Precise mathematical comparisons are not possible, because
these 4- and 8-month seasons are not directly comparable to the 2–3-month
timescale of the young water fractions Fyw* and
PFyw*, and also because these young
water fractions are annual averages, whereas the fs and ηs pertain
to individual seasons. Nonetheless, all of these lines of evidence imply
that significant fractions of streamflow must originate from precipitation
in previous seasons and conversely that significant fractions of
precipitation become streamflow in future seasons. This in turn implies
significant water storage within the catchment, either as snowpack or as
groundwater.
Comparison with new water fractions estimated by ensemble hydrograph
separation
Another approach for quantifying timescales of storage and transport using
isotopic tracers is ensemble hydrograph separation. Ensemble hydrograph
separation uses the regression slope between tracer fluctuations in
streamwater and precipitation to quantify the “new water fraction”, the
average fraction of streamflow that is “new” since the previous
precipitation sample (Kirchner, 2019a). Thus, in this case, because the
precipitation isotopes are averaged over a roughly 2-week sampling
interval, the new water fraction quantifies the fraction of streamflow that
is younger than about 2 weeks. This biweekly new water fraction,
QFnew, can be estimated from the regression
slope parameter β in the linear regression equation
yj=βxj+α+εj,withyj=δQj-δQj-1andxj=δ‾Pj-δQj-1,
where δ‾Pj and δQj are the isotope signatures in precipitation and
streamflow, respectively, in the jth sampling interval (and where the
overbar on δ‾Pj indicates that it is an average
over that interval), and the regression intercept α and error term
εj subsume any bias or random error introduced by
fractionation, measurement noise, and so forth (Kirchner, 2019a). If many
sampling intervals have no precipitation, one must account for the number of
intervals with precipitation, as a fraction of the total (see Kirchner, 2019a,
for details), but here we can overlook this because nearly every 2-week
interval at Hubbard Brook has precipitation. Weighting the regression in Eq. (32) by discharge yields the volume-weighted new water fraction of
streamflow, QFnew*.
Uncertainty estimates for QFnew* and similar volume-weighted quantities should take account of the
reduced degrees of freedom that result from the uneven weighting, as
described in Eq. (19) of Kirchner (2019a).
Following the approach of Eq. (4), we can multiply QFnew* by the ratio of mean discharge to mean
precipitation to obtain the volume-weighted new water fraction of
precipitation PFnew*, the fraction of
precipitation that, on average, leaves the catchment as streamflow within
the sampling interval (in this case, 2 weeks):
PFnew*=QFnew*Q‾P‾.
In the language of Sect. 1, Eq. (33) splits the precipitation end-member into two
fractions: the average fraction that leaves as streamflow within the
sampling interval (PFnew*) and the
average fraction that does not (1-PFnew*). For this reason, PFnew* can also
be termed a “forward” new water fraction because it divides precipitation
into two different future fates. Likewise QFnew* can be termed a “backward” new water
fraction because it divides streamflow according to its origins as
precipitation in the recent or distant past. In contrast to end-member
mixing and end-member splitting, this approach is based on correlations
between tracer fluctuations in streamflow and precipitation, rather than
mass balances. Thus it can be applied even if the underlying tracer time
series are incomplete.
Applying this approach to the Hubbard Brook record, and using the total
discharge in each sampling interval as weights, we estimate the
volume-weighted biweekly new water fraction of discharge QFnew* as 8.3±1.9 % and the
corresponding volume-weighted biweekly new water fraction of precipitation
PFnew* as 5.3±1.2 %. These
results mean that, on average, about 5 % of precipitation leaves
the catchment as streamflow in the following 2 weeks, and this makes up
about 8 % of streamflow.
One can also apply this regression approach to subsets of the data,
highlighting time periods or catchment conditions of particular interest
(Kirchner, 2019a). For comparison with the results presented in Sects. 2.4
and 2.6 above, we divided the time series into four seasons: the 4-month
snowy season (December–March), the 4-month growing season
(June–September), and the 2-month spring and fall seasons in between
(April–May and October–November, respectively). The volume-weighted
regressions for these four seasons (Fig. 10) show that tracer fluctuations
in precipitation and streamflow are weakly correlated during the snowy
season (Fig. 10a), much more strongly correlated in the spring (Fig. 10b),
and correlated to an intermediate degree during the growing season and the
fall (Fig. 10c–d). The volume-weighted biweekly new water fraction of
discharge QFnew* is zero
within error (2.2±3.3 %) during the snowy season (Fig. 10a), even
though at the 4-month seasonal timescale (Fig. 4), roughly half of
snowy-season streamflow originates as snowy-season precipitation. Considered
together, these results would seem to imply that almost all winter
precipitation is stored in the catchment for at least 2 weeks (as either
snowpack or subsurface storage), effectively decoupling precipitation and
streamflow on that timescale, but roughly half eventually melts or seeps out
to streams sometime during the winter.
Ensemble hydrograph separation using biweekly isotope
measurements at Hubbard Brook Watershed 3. Straight lines show least-squares
regressions weighted by cumulative stream discharge over each 2-week
sampling interval. Curved lines indicate 95 % confidence bounds for the
fits. The regression slopes yield ensemble estimates of the biweekly
volume-weighted new water fraction of discharge (the volume fraction of
discharge that originated from precipitation that fell in the previous
2-week sampling interval); QFnew*=0.022±0.033
during the snowy season (December–March, panel a), 0.220±0.078 during
the spring (April and May, panel b), 0.106±0.028 during the growing
season (June–September, panel c), and 0.116±0.034 during the fall
(October and November, panel d). Rescaling these biweekly event new water
fractions by the ratio between seasonal discharge and seasonal precipitation
yields the biweekly volume-weighted new water fractions of precipitation
(the volume fraction of precipitation that leaves as discharge within the
following 2-week sampling interval); PFnew*=0.015±0.022
during the snowy season, 0.311±0.111 during the spring, 0.027±0.007 during the growing season, and 0.076±0.023 during the fall.
Axes vary from panel to panel, but their ratios are held constant, so the
plotted lines correctly depict the relative steepness of the regression
slopes.
During the growing season (Fig. 10c), the volume-weighted biweekly new water
fraction of discharge QFnew*
is 10.6±2.8 %. This fraction is small enough to be broadly
consistent with the observation that, on a seasonal timescale, about half of
growing-season streamflow originates as growing-season precipitation (Fig. 7), although an exact equivalence is difficult to draw because the fraction
of “new” water in streamflow declines over time following each event.
(Nonetheless, if, for example, the fraction of streamflow less than 2 weeks
old were similar to, or even larger than, the fraction less than 4 months
old, that would indicate a clear problem with one or both estimates.) During
the fall the biweekly new water fraction is similar (11.6±3.4 %),
but during the spring it is distinctly higher (22.0±7.8 %),
presumably due to more saturated catchment conditions.
The biweekly new water fractions of precipitation PFnew* yield further insights. The biweekly new water
fraction of precipitation is markedly higher during the spring (31.1±11.1 %), reflecting greater transmission of new water to streamflow under
wet catchment conditions. By contrast, very little precipitation is
transmitted to streamflow on a 2-week time frame during either the snowy
season (1.5±2.2 %) or the growing season (2.7±0.7 %),
reflecting the fact that there is relatively little streamflow of any kind
during those periods. In the snowy season this is due to snowpack storage;
in the growing season it is due to evapotranspiration. The essential
difference between the two is that the snowpack episodically melts, with the
result that about one-fourth of snowy-season precipitation eventually
becomes snowy-season streamflow (Fig. 4), whereas the evapotranspired
water is lost forever, with the result that only about 10 % of
growing-season precipitation eventually becomes growing-season streamflow
(Fig. 7).
Figure 11 shows the same ensemble hydrograph separation approach, applied
separately to each month of the year. The volume-weighted biweekly new water
fraction of discharge QFnew*
is lowest in January and February (when temperatures at Hubbard Brook are
the coldest) and peaks during snowmelt in April. The rest of the year it
hovers around 10 %. The volume-weighted biweekly new water fraction of
precipitation PFnew* is zero within
error from January through March, then abruptly rises to 43±25 %
during April, declines to 2 % or less throughout the growing season from
June through September, and then rises to 5 %–9 % until the end of the year.
Here again we see the effects of winter freezing and summer
evapotranspiration in limiting streamflow (as well as recent contributions
of precipitation to it). We also see the effects of catchment wetness during
snowmelt facilitating the transmission of large fractions of recent
precipitation to streamflow as well as the increase in precipitation
reaching the stream from October through December, following the cessation
of the growing season. This analysis provides striking evidence that during
about half of the year, in mid-summer and mid-winter, nearly no
precipitation reaches the stream during the first 2 weeks after it falls.
More generally, this analysis also demonstrates that ensemble hydrograph
separation can yield useful insights into the partitioning of precipitation
into prompt and more distant streamflow, even based on biweekly tracer data.
Furthermore, this analysis shows that new water fractions of precipitation
can be combined with end-member splitting analyses to gain insight into
evapotranspiration and subsurface storage as controls on how much recent
precipitation reaches streams.
Seasonal patterns in (a) average precipitation and streamflow
fluxes, (b) biweekly volume-weighted new water fractions of streamflow
QFnew* (fraction of streamflow
derived from precipitation that fell in the previous 2 weeks), and (c)
biweekly volume-weighted new water fractions of precipitation
PFnew* (fraction of
precipitation that becomes streamflow within the following 2 weeks), as
determined from ensemble hydrograph separation (Eqs. 32 and 33; Fig. 10).
Dashed lines in (b) and (c) indicate new water fractions of zero. Average
precipitation fluxes (a) vary little from month to month, whereas average
streamflow fluxes show clear high flows resulting from snowmelt from March
through May and clear low flows attributable to evapotranspiration losses
from July through September. Both intervals are marked by gray shading.
Ensemble hydrograph separations imply that recent (previous 2 weeks)
precipitation comprises about 20 % of streamflow during the snowmelt peak
in April, roughly 0 % (within error) during the cold winter months of
January, February, and March, and roughly 10 % (within error) during the
rest of the year. These streamflow fractions can be re-expressed as
fractions of precipitation by multiplying by monthly streamflow and
dividing by monthly precipitation. The resulting biweekly new water
fractions of precipitation quantify the fractions of precipitation that
leave the catchment as streamflow within the following 2 weeks (c). These
are zero within error in January, February, and March, rise to 43 % during
April snowmelt, decline to 2 % or less throughout the growing season (June
through September), and then rise to 5 %–9 % during October, November, and
December.
Assumptions, limitations, and applicationsFundamental assumptions
Many of the assumptions underlying end-member splitting are the same as
those that underlie end-member mixing. End-member mixing requires,
fundamentally, that there are only two end-members (if we have one tracer),
or n+1 end-members (if we have n non-redundant tracers), that contribute to
the measured mixture(s). More crucially, end-member mixing requires that
these are the only end-members (in the real world, not just the only end-members in your
theory, your model, or your sampling program). This assumption is broadly
met by our two end-members, because precipitation is the ultimate source of
catchment streamflow and evapotranspiration (assuming other inputs such as
groundwater inflows, condensation, or fog deposition are trivial by
comparison), and because we have divided annual precipitation into two
seasons, without gaps or overlaps.
End-member mixing also requires that the tracer signatures of the
end-members and mixture(s) have been measured without bias. This assumption
is broadly met, in our case, by measuring the volume-weighted average
isotope signatures of precipitation and streamflow and measuring them for
long enough that carryover effects at the beginning and end of the period
are likely to be small. However, one must also be aware of possible isotopic
fractionation in the precipitation sampler itself. It is also possible that
an unbiased sample of precipitation could nonetheless be a biased sample of
the precipitation that actually becomes streamflow. If, for example,
lower-intensity precipitation events tend to be isotopically heavier
(Dansgaard, 1964) and more likely to be lost to canopy interception, an
unbiased sample of precipitation will be isotopically heavier than the
precipitation that eventually flows through the catchment and becomes
streamflow. This in turn would lead to an underestimate of summer
precipitation (and an underestimate of winter precipitation) as contributors
to streamflow.
Lastly, end-member mixing requires that the tracer signatures of the fluxes
connecting the end-members to the mixture(s) are not substantially altered
by fractionation (i.e., tracer-selective addition or removal of water). For
example, although evaporation fluxes are likely to be strongly fractionated,
if the waters that are left behind eventually evaporate completely (as may
often occur during canopy interception, for example; Allen et al., 2017),
the remaining precipitation that eventually becomes streamflow may not be
substantially fractionated. Streamwater at Hubbard Brook lies close to the
local meteoric water line (Fig. 2b), suggesting that any such fractionation
effects are likely to be small. In other settings, such as conifer forests
in arid climates, one might expect greater evaporation/sublimation of
intercepted rain and snow, along with the resulting fractionation of the
remaining waters. For this reason, in Sect. 3.3 below, we quantify how
different types of fractionation would affect our analysis.
In addition to the assumptions outlined above for end-member mixing,
end-member splitting additionally requires that the sampled mixture(s)
represent all of the outputs from the system except one, and that the water
fluxes in these all-but-one outputs, as well as the end-members, can be
quantified with reasonable accuracy. One can see from Eqs. (4), (22)–(24),
and (31) that uncertainties in these water fluxes will propagate
proportionally through to uncertainties in the end-member splitting
fractions. In addition, calculating the end-member mixing fractions of
evapotranspiration fluxes (Eqs. 8 and 18) requires that the other inputs and
outputs are known precisely enough that ET can be calculated with sufficient
accuracy by mass balance. Our proof-of-concept demonstration at Hubbard
Brook is facilitated not just by the availability of isotope data, but also
by a reliable long-term catchment water balance.
Sensitivity to errors in mass fluxes
End-member mixing calculations are not based on mass flux measurements and
therefore are independent of errors in mass fluxes (except to the extent
that they are needed to accurately estimate volume-weighted tracer
signatures for the end-members and mixtures). End-member splitting
calculations, on the other hand, require mass flux measurements and thus are
potentially vulnerable to errors in them. We can straightforwardly calculate
the sensitivity of these calculations to mass flux errors by (for example)
differentiating Eq. (22) by its two component fluxes:
∂ηPs→Qs∂Qs=1Psδ‾Qs-δ‾Pwδ‾Ps-δ‾Pw=ηPs→QsQsor∂ηPs→QsηPs→Qs=∂QsQs
and
∂ηPs→Qs∂Ps=-QsPs2δ‾Qs-δ‾Pwδ‾Ps-δ‾Pw=-ηPs→QsPsor∂ηPs→QsηPs→Qs=-∂PsPs.
Equations (34)–(35) show that an x percent overestimate in
Qs would lead, all else equal, to an x percent overestimate
in the end-member splitting fraction ηPs→Qs, and that an x
percent overestimate in Ps would lead, all else equal, to an
x percent underestimate in ηPs→Qs. Equation (35) assumes that x is small; if that
is not the case, one can directly simulate the effect of large errors in
Ps by solving Eq. (22) for a range of Ps values.
We can similarly differentiate Eq. (18) by its three component fluxes to
quantify how flux measurement errors would affect estimates of the fraction
of ET originating as summer precipitation, fET←Ps:
∂fET←Ps∂Ps=1-fET←PsET,∂fET←Ps∂Pw=-fET←PsET,and∂fET←Ps∂Q=fQ←Ps-fET←PsET.
Figure 12a shows how errors in the water fluxes Ps,
Pw, and Q at Watershed 3 would alter the estimates of
fET←Ps and ηPs→ET shown in Fig. 4. As one can see
from Fig. 12a, fET←Ps is least
sensitive to errors in Ps (solid light blue curve); this is
because Ps appears in both the numerator and denominator of
Eq. (18), with mostly offsetting effects. Although Q also appears in both the
numerator and denominator, in the numerator it is multiplied by
fQ←Ps so errors in Q will not
have such cleanly offsetting effects (dashed light blue curve). Errors in
Pw (dotted light blue curve) are the most consequential
because Pw appears only in the denominator of Eq. (18).
Readers will note that sufficiently severe flux measurement errors can lead
to calculated values of fET←Ps
that exceed 1; this nonphysical result can arise when the water fluxes and
tracer signatures in Eq. (18) become sufficiently inconsistent with one
another.
Sensitivity of end-member splitting fractions to measurement
errors in water fluxes (a) and tracer signatures (b, c). Light blue curves
show variations in the fraction of evapotranspiration
(fET←Ps; a, b) and summer streamflow
(fQs←Ps; c) that originates as summer
precipitation. Dark blue curves show variations in the fraction of summer
precipitation that eventually evapotranspires (ηPs→ET; a, b)
or becomes summer streamflow (ηPs→Qs; c). Solid curves show effects
of errors in Ps(a) and δ‾Ps(b, c). Dotted curves show
effects of errors in Pw(a) and δ‾Pw(b, c). Dashed curves show
effects of errors in Q(a), δ‾Q(b), and δ‾Qs(c).
Curves are calculated using Eqs. (11), (18), (22), and (25), using input
values from Fig. 4, adjusted as shown on the x axis of each panel.
Potential effects of isotopic fractionation
End-member splitting, just like end-member mixing, is potentially vulnerable
to the effects of isotopic fractionation. If, for example, a fraction of
precipitation evaporates from the rainfall collector, the remaining water,
which will be sampled and analyzed, will be isotopically heavier than the
precipitation that it is supposed to represent. Alternatively, if the
precipitation samples themselves are not isotopically fractionated, but the
precipitation that enters the catchment is fractionated before it becomes
streamflow, then the sampled precipitation will be isotopically lighter than
the precipitation that it is supposed to represent (i.e., the precipitation
that eventually becomes part of streamflow). How much the precipitation that
reaches the stream is fractionated will depend, not only on how much
evaporates and on ambient temperature and humidity under which that
evaporation occurs, but also on how much the evaporating waters are mixed
with (or separated from) the waters that are left behind (Brooks et al.,
2010; Sprenger et al., 2016). To the extent that the evaporating waters are
separated from those that ultimately reach the stream, their isotopic
fractionation will not be reflected in the streamflow isotope signature. An
example of such a process is canopy interception; if the intercepted
precipitation mostly evaporates after the rain has stopped, and evaporates
completely, it leaves no isotopic signal in the water that reaches the
stream (Gat and Tzur, 1967; Allen et al., 2017). Alternatively, if the
evaporation flux comes from a well-mixed pool that also supplies streamflow,
that streamflow will bear the isotopic fingerprint of evaporative
fractionation, with streamflow falling below the local meteoric water line
on a dual-isotope plot. In any case, a benefit of using stream water to
infer the seasonal origins of evapotranspired waters is that fractionation
effects should be much smaller than they would be in sampled xylem or soil
water, for which evaporation effects must be compensated to infer their
seasonal origins (Benettin et al., 2018; Bowen et al., 2018; Allen et al.,
2019a).
One can straightforwardly estimate how isotopic fractionation would affect
end-member mixing and splitting fractions by differentiating the
corresponding equations by the corresponding input isotope values. For
example, we can differentiate Eq. (11) by its three isotopic inputs to
quantify how isotopic fractionation could alter estimates of
fQs←Ps, the
fraction of summer streamflow that originates as summer precipitation:
∂fQs←Ps∂δ‾Ps=-fQs←Psδ‾Ps-δ‾Pw,∂fQs←Ps∂δ‾Pw=fQs←Ps-1δ‾Ps-δ‾Pw,and∂fQs←Ps∂δ‾Qs=1δ‾Ps-δ‾Pw,
where fQs←Ps=δ‾Qs-δ‾Pwδ‾Ps-δ‾Pw-1. The fraction of summer precipitation that eventually
becomes summer streamflow, ηPs→Qs, equals fQs←Ps rescaled by Qs/Ps, the
ratio of summer streamflow to summer precipitation (Eq. 22), so the effects
of isotopic fractionation on ηPs→Qs are likewise proportional to those derived
directly above for fQs←Ps,
∂ηPs→Qs∂δ‾Ps=-ηPs→Qsδ‾Ps-δ‾Pw,∂ηPs→Qs∂δ‾Pw=ηPs→Qs-Qs/Psδ‾Ps-δ‾Pw,and∂ηPs→Qs∂δ‾Qs=Qs/Psδ‾Ps-δ‾Pw.
As another example, we can differentiate Eq. (18) by its three isotopic
inputs to quantify how isotopic fractionation could alter estimates of
fET←Ps, the fraction of
evapotranspiration that originates as summer precipitation:
∂fET←Ps∂δ‾Ps=QETfQ←Psδ‾Ps-δ‾Pw,∂fET←Ps∂δ‾Pw=QET1-fQ←Psδ‾Ps-δ‾Pw,and∂fET←Ps∂δ‾Q=QET-1δ‾Ps-δ‾Pw,
where fQ←Ps=δ‾Q-δ‾Pwδ‾Ps-δ‾Pw-1. Rescaling
fET←Ps by ET /Ps,
the ratio of evapotranspiration to summer precipitation, yields ηPs→ET, the fraction of summer
precipitation that eventually evapotranspires (Eq. 25), so we can calculate the
effects of isotopic fractionation on ηPs→ET by rescaling Eq. (39) by the same
ratio:
∂ηPs→ET∂δ‾Ps=QPsfQ←Psδ‾Ps-δ‾Pw,∂ηPs→ET∂δ‾Pw=QPs1-fQ←Psδ‾Ps-δ‾Pw,and∂ηPs→ET∂δ‾Q=QPs-1δ‾Ps-δ‾Pw.
Equations (37)–(40) show that, perhaps counterintuitively, if both summer
and winter precipitation are fractionated in the same direction, their
effects reinforce one another rather than tend to cancel each other out;
their terms have the same signs in each of the four equations. For example,
an overestimate of δ‾Ps in Eq. (37)
will lead to an underestimate of fQs←Ps, because a larger δ‾Ps will increase the denominator of
fQs←Ps (see Eq. 11). However, an overestimate of δ‾Pw will also lead to an underestimate
of fQs←Ps, because
the numerator of fQs←Ps will always be smaller than the denominator (since
the fraction f must be less than 1), so a larger δ‾Pw will shrink the numerator of
fQs←Ps more than
the denominator in percentage terms.
Figure 12 demonstrates how calculations of
fQs←Ps,
fET←Ps, ηPs→Qs, and ηPs→ET would be affected by errors in
the mass fluxes and isotope signatures that they use as inputs. Figure 12b
and c show that errors in δ‾Ps (solid
lines) and δ‾Pw (dotted lines)
reinforce, rather than offset, one another, but that they both would tend to
be counteracted by errors in δ‾Q (dashed lines),
assuming that these errors all have the same sign. Figure 12 is based on
input values from Fig. 4; for other input values the results would differ in
detail, but we expect the overall patterns to be similar.
Potential applications
These methods may provide new insight into how climate change could affect
terrestrial ecosystems and water resources. Climate change projections
typically involve precipitation increases or decreases in specific seasons,
and the tools presented here provide empirical insights into how different
seasons' precipitation is partitioned into evapotranspiration or streamflow.
At Hubbard Brook Watershed 3, for example, only a small fraction of
snowy-season precipitation is evapotranspired (Fig. 4), and a large fraction
of evapotranspiration is derived from precipitation that falls during the
growing season itself (Fig. 7). These results suggest that tree-ring
cellulose is likely to record the isotopic signatures of summer
precipitation, rather than those of mean annual precipitation. These results
also suggest that forest growth at Hubbard Brook is likely to be sensitive
to changes in growing-season precipitation, but less sensitive to changes in
winter snowfall. By contrast, roughly half of growing-season streamflow at
Watershed 3 originates as precipitation outside of the growing season (Fig. 7), suggesting that summer streamflow could be strongly affected by changes
in precipitation in other seasons.
Hypotheses such as these could be tested using isotope records that
encompass multiple years with contrasting climates. We could, for example,
separate such a long-term record into years with above-average and
below-average winter precipitation (or growing-season rainfall). We could
then examine how the seasonal partitioning of precipitation, and the
seasonal origins of streamflow and evapotranspiration, differed between
these different sets of years. If, for example, evapotranspiration fluxes in
drier summers are accompanied by smaller contributions from summer
precipitation and greater contributions from winter precipitation (smaller
fET←Ps and larger
fET←Pw), then winter
precipitation may be able to buffer the effects of shifts in summer
precipitation on forest growth. Conversely, the lack of such a compensatory
response would suggest greater vulnerability of forest growth to changes in
summer precipitation. Through such analyses (of which one is underway), we
can transition from asking “which seasons' water do ecosystems use?” to
asking “which seasons' water do they depend on?”.
End-member splitting may also help in illuminating hydrological transport,
storage, and mixing processes. For example, if substantial fractions of
summer precipitation become summer streamflow despite widespread
soil-moisture deficits throughout the catchment (which is not the case at
Hubbard Brook), this would indicate that summer precipitation can bypass the
soil via preferential flow, contrary to the common model representation of
soils as well-mixed “buckets”. Such a scenario could explain why trees
throughout much of Switzerland were recently found to be predominantly using
winter precipitation in mid-summer of 2015, despite enough summer
precipitation having fallen to saturate soils to their median rooting depths
(Allen et al., 2019a). By contrast, however, streamwater isotopes in a
network of Swiss catchments imply that roughly equal fractions of winter and
summer precipitation typically become streamflow (Allen et al., 2019b),
suggesting that the relatively dry summer of 2015 may have made the
trees more reliant than usual on water from winter precipitation. This
example illustrates the potential of combining end-member splitting analysis
with direct isotopic sampling of xylem water and soil water.
The relative amounts of precipitation becoming same-season streamflow or ET
vs. “crossing over” to become streamflow or ET in other seasons also
provide constraints on the shapes of the transit time distributions of the
precipitation that becomes streamflow and of the precipitation that
evapotranspires. End-member splitting may also be helpful for model
calibration, validation, and testing, because it provides different
information than is provided by hydrometric input/output data. Unlike direct
tests against isotopic time series, end-member splitting analysis provides a
“fingerprint” or “signature” of catchment behavior for models to be tested
against, an approach that will often have greater diagnostic power (Kirchner
et al., 1996). End-member splitting also provides spatially and temporally
integrated information, in contrast to point measurements of xylem and soil
water, which cannot be readily generalized to the scales of most hydrologic
models. Furthermore, because end-member splitting analysis can be performed
with relatively short weekly or biweekly time series, it can potentially be
applied in a wide range of sites where only low-frequency isotopic data are
available, rather than the few sites where direct model calibration and
testing against isotope time series would be feasible.
The analyses presented in Sect. 2, as well as the potential applications
outlined in this section, have focused on the coupling of precipitation to
streamflow and evapotranspiration within and between seasons. In temperate
climates and continental interiors, such analyses are facilitated by the
strong seasonal cycle that is typically found in the isotopic composition of
precipitation. All of the approaches presented here require that
precipitation can be separated into two seasons that are isotopically
distinct. This will not be possible in all cases. Exceptions include coastal
or tropical sites lacking strong seasonality in precipitation isotopes and
Mediterranean climates in which almost all precipitation falls within a
single season.
Such cases where precipitation isotope seasonality is weak or absent present
intractable problems for seasonally oriented analyses, but also present
opportunities for analyses based on isotopic differences between other
groupings of precipitation events. In field settings spanning large
elevation gradients, one could potentially use the isotopic variation in
precipitation with altitude (the “altitude effect”; Dansgaard, 1954, 1964;
Siegenthaler and Oeschger, 1980), within an end-member splitting framework,
to contrast the fates of precipitation falling in the higher vs.
lower parts of a river basin. Alternatively, one could potentially make use
of the fact that low-intensity precipitation is often isotopically heavier
than high-intensity precipitation, due to greater isotopic fractionation of
raindrops as they fall (the “amount effect”; Dansgaard, 1964). Where the
contrast between low-intensity and high-intensity storms is the dominant
source of variability in precipitation isotopes (e.g., in some tropical
regions; Jasechko and Taylor, 2015), end-member splitting analysis could be
used to contrast the fates of low-intensity and high-intensity
precipitation, providing new insight into transport, storage, and runoff
generation at the catchment scale. As an extreme example of contrasting
storm intensities, one could potentially use tropical cyclones and all other
precipitation as the two end-members, because tropical cyclones are
isotopically much lighter than any other tropical precipitation (Lawrence
and Gedzelman, 1996).
Concluding remarks
We make no particular claim for the novelty of the approach we have outlined
here, since it represents a conceptually straightforward combination of
end-member mixing and isotope mass balance methods, both of which are well
established. End-member splitting is nonetheless noteworthy because it
represents a different perspective. It invites questions that are seldom
asked, such as “where does precipitation go?” (rather than “where does
streamflow come from?”), and provides a framework for answering them. Such
questions have previously been approached through simulation models (e.g.,
Benettin et al., 2015, 2017), but end-member splitting
provides a model-independent way to answer them directly from data.
The analyses presented in Sect. 2 above serve both as a worked example
showing how end-member splitting can be applied in practice and as a
proof-of-concept study that illustrates its potential. The techniques
outlined in Sect. 2 can be used to determine the seasonal origins of
streamflow (Sect. 2.2) and evapotranspiration (Sect. 2.3) as well as the
seasonal partitioning of precipitation into evapotranspiration and
streamflow (Sect. 2.4). We also show that one can infer how the seasonal
origins of streamflow shift from month to month and conversely how
precipitation is partitioned among monthly streamflows (Sect. 2.5).
Here we have analyzed Hubbard Brook Watershed 3 as a test case. The results
illustrate how end-member mixing and splitting yield different insights,
which together give a more complete picture of catchment behavior. At
Watershed 3, for example, almost all evapotranspiration is derived from
rainy-season precipitation, but only about half of rainy-season
precipitation eventually transpires (Fig. 4). One sixth of rainy-season
precipitation is eventually discharged during the snowy season, but this
accounts for half of snowy-season streamflow (Fig. 4). Only about 10 % of
growing-season precipitation becomes discharge during the growing season,
but this accounts for nearly half of growing-season streamflow (Fig. 7). The
other half of growing-season streamflow is derived from just 7 % of
dormant-season precipitation (Fig. 7). The largest discharges of
rainy-season precipitation occur during snowmelt, when rainy-season
precipitation makes up the smallest fraction of streamflow; conversely, the
smallest discharges of rainy-season precipitation occur during the growing
season, when it makes up the largest fraction of streamflow (Fig. 6). In all
the cases shown here (Figs. 4, 7, and 8), a substantial fraction of each
season's streamflow originates as precipitation in other seasons. These
results therefore imply substantial inter-seasonal catchment storage, in
either snowpacks or groundwaters.
Code and data availability
R scripts that perform the main calculations described in this paper, along
with demonstration input data and output files, are available from the
EnviDat repository (10.16904/envidat.91, Kirchner, 2019b). The source data
used in this paper are available from the cited references.
The supplement related to this article is available online at: https://doi.org/10.5194/hess-24-17-2020-supplement.
Author contributions
JWK and STA jointly developed the end-member splitting approach. JWK
performed the analysis presented here and drafted the paper. Both authors
discussed all aspects of the work and jointly edited the manuscript.
Competing interests
The authors declare that they have no conflict of interest.
Special issue statement
This article is part of the special issue “Water, isotope and solute fluxes in the soil–plant–atmosphere interface: investigations from the canopy to the root zone”. It is not associated with a conference.
Acknowledgements
We thank the Hubbard Brook Ecosystem Study, and particularly Mark Green and
John Campbell, for making the data that we used in our analysis publicly
available. This analysis is a by-product of research in forest water use
sponsored by the Swiss Federal Office of the Environment.
Review statement
This paper was edited by Lixin Wang and reviewed by Pertti Ala-aho and Sylvain Kuppel.
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