Stephen P. Good, David Noone, Gabriel Bowen correspondence to: stephen.good@oregonstate.edu
This PDF file includes:
Materials and Methods
Figs. S1 to S3
References (31-37)
1

Materials and Methods
Local HDO budget:
We propose a framework for catchment isotope hydrology that maximizes the utility of both runoff and evapotranspiration isotope ratio data while also reconciling two prior flux partitioning approaches. Within this framework, we define two distinct water pools: soil/bound waters and surface/mobile waters. For the bound water, the mass balance of
H
2
O and HDO over a finite time dt are
1a B
0
+ Pdt = ( B
0
+ dB ) + Idt + Tdt + E
B dt + Ldt ,
1b B
0
R
B
+ PR
P dt = ( R
B
+ dR
B
) B
0
+ IR
P dt + TR
T dt + E
B
R
E ( B ) dt + LR
L dt , while for the mobile waters the mass balance for H
2
O and HDO are
2a M
0
+ Ldt = ( M
0
+ dM ) + E
M dt + Qdt ,
2b M
0
R
M
+ LR
L dt = ( R
M
+ dR
M
) M
0
+ E
M
R
E ( M ) dt + QR
Q dt .
In equations 1 and 2, the bulk fluxes of precipitation ( P ), interception ( I ), transpiration
( T ), soil evaporation ( E evaporation ( E
M
B
), soil leakage ( L ) from the bound to mobile pools, surface water
), and runoff ( Q ) out of the catchment, are each associated with an isotope ratio ( R i
) where i is the appropriate flux. For both bound and mobile water the initial amount of HDO in the catchment is given by the bulk water amount ( B
0 and M
0 respectively) multiplied by its isotopic ratio ( precipitation that is not intercepted (1 - f
I evaporated ( f
E(B)
=E
(B)
/P , with isotope ratio R
R
B
E(B)
and R
M
respectively). The fraction of
= 1 - I / P , with an isotope ratio R
P bound water pool where some fraction is either transpired ( f
T
=T/P
) passes into the
, with isotope ratio R
) or leaks out to the mobile water pool
T
),
( f
L
=L/P , with isotope ratio R
L
). Water that has entered the mobile water pool then either evaporates ( f
E(M)
=E
(M)
/P , with isotope ratio R
EM
) or leaves the catchment as surface runoff ( f
Q
=Q/P , with isotope ratio R
Q
).
Bound waters held in soils are subjected to both transpiration and evaporation as well as leakage into mobile/surface waters. Over extended timescales, the isotopic composition of transpired water is unfractionated relative to soil waters ( 13 ) and therefore assumed equivalent to the bound water pool as
3 R
T
= R
B
.
The isotopic composition of water evaporated from soils is estimated based on the Craig-
Gordon model ( 14 ) as
4 R
E ( B )
= α
K
α
*
R
B
− hR
A
1 − h
= α
CG 1
R
B
+ α
CG 2
.
Where R
A
is the isotopic composition of the surface atmosphere, α
*
is the equilibrium vapor to liquid fractionation factor ( 31 ) and h is the vapor pressure relative to the saturation value at the temperature of the evaporating surface. Placeholder fractionation coefficients α
CG1
= α
*
α
K
/(1h ) and α
CG2
= α
K hR
A
/(1h ) are used to simplify equation 4 equation for further manipulation. The kinetic fractionation factor, α
K
= C
K
(1h ), where
C
K ranges from 12.5 to 2 depending on surface conditions and diffusive path length ( 32 ).
Finally, the connectivity of waters within the soil columns to surface water determines
2

the isotopic composition of water entering the mobile pool. We represent this mixture of possible catchment water flowpaths as
5 R
L
= cR
B
+ (1 − c ) R
P
, where c is a parameter we term the ‘hydrologic connectivity’ of the system. In a fully connected system ( c= 1) water accessible to plants and possibly subjected to soil evaporation also moves into streams. In a disconnected ( c =0) system characterized by preferential flow, soil waters do not interact with surface waters and therefore water entering streams and rivers has an isotopic composition equivalent to that of rainfall.
Equations 3, 4, and 5 are substituted into 1, which is integrated and solved for the isotopic ratio of bound water at steady state (i.e. the asymptotic limit at large t ) as
6 R
B
=
(1 − f
I
− (1 − c cf
L
+ f
T
) f
L
) R
P
− f
E ( B )
α
CG 2 .
+ α
CG 1 f
E ( B )
Our application of this framework assumes steady-state conditions are achieved at the yearly timescale for the isotopic composition of bound and mobile waters. However input precipitation and output evaporation fluxes vary throughout the year. Flux weighted input and output values are calculated based on the distribution of rainfall and evapotranspiration fluxes throughout the year as well as seasonal variation in meteorological conditions. For precipitation this takes the form of
7 R
P
=
12
m = 1
R
P
12
( m ) P ( m )
m = 1
P ( m )
, while for evaporation fluxes, the placeholder fractionation factors, α
CG1 weighted based on monthly meteorological conditions as
and α
CG2
, are
8a α
CG 1
=
12
∑ m = 1
α
*
( m ) α
K
1 − h ( m )
( m )
12
∑ m = 1
ET ( m )
ET ( m )
,
12
∑ m = 1
α
K
( m ) h ( m ) R
A
1 − h ( m )
( m )
ET ( m )
8b α
CG 2
= ,
12
∑ m = 1
ET ( m )
In equations 7 and 8, m is the month of the year and ET=I+T+E evapotranspiration flux. In this approach, the values of R
B
B
+E
M
and R
M and R
Q
,
is the and therefore also
, represent the mean annual value. When there is no soil evaporation ( f
E(B)
=0),
R bound water is unfractionated and its isotopic composition equal to that of rainfall, with
T equation 6 simplifying to R
B
=R
P
.
Similarly, when all bound output flux consists of evaporation from soils, f
T
= f
L
=0, under arid conditions, h =0, then equation 6 simplifies to
3

R
B
= R
P
/ α
K
α
*
. Finally, when no evaporation is possible, h =1 , bound water is in equilibrium with atmospheric vapor, R
B
= R
A
/ α
*
.
For surface waters within a catchment, we assume mobile waters are well mixed and runoff exiting a catchment is equal to the isotopic composition of mobile water
9 R
Q
= R
M
.
The evaporation flux from the mobile water pool is fractionated following Craig-Gordon
( 14 ) in a similar manner as above,
10 R
E ( M )
= α
K
α
*
R
M
− hR
A
1 − h
= α
CG 1
R
M
+ α
CG 2
.
Equations 9 and 10 are then substituted into 2, which is integrated and solved for the isotopic ratio of the surface water pool at steady state
11 R
M
= f
L
R
L
− α
CG 2 f
E ( M ) . f
Q
+ α
CG 1 f
E ( M )
Note that runoff based partitioning approaches, e.g. ( 3 ), may be represented by setting f
E(B)
=0 and c= 0 in equation 6. Typical micro-meteorological approaches do not consider surface water evaporation as this flux often occurs beyond the flux footprint, and thus f
L
= 0 for these studies. In our analysis, runoff leaving each cell is routed directly to the ocean and no upstream contributions are considered. Finally, the total surface to atmosphere flux isotopic composition is the weighted sum of interception, transpiration, soil evaporation, and surface water evaporation, calculated as
12 R
ET
= f
I
R
P
+ f
T
R
B
+ f
E ( B )
R
E ( B )
+ f
E ( M )
+ f
E ( B )
+ f
E ( M )
R
E ( M ) . f
I
+ f
T
When implementing the above local isotope hydrology framework, if the hydrologic flux partitioning ( f
I
, f
Q
, f
T
, f
E(B)
, and f
E(M)
) is known and the isotope ratio of input precipitation
( R
P
) is specified, the isotope ratio of output evapotranspiration ( R
ET
) and runoff ( R easily determined. Conversely, if the input and output isotope ratios ( R
P
Q
) are
, R
Q
, and R
ET
) are known, and total runoff and interception are specified, the remaining hydrologic fractions of f
T
, f
E(B)
, and f
E(M)
are easily determined. This second, inverse approach is applied here to partition input precipitation into output fluxes.
Global HDO budget:
Because local evapotranspiration fluxes, and to a lesser extent, runoff isotope composition are poorly quantified, we apply the above framework globally by resolving the global HDO isotope ratios in total continental runoff and evapotranspiration and comparing these values to those obtained independently from a HDO mass balance of the oceans and atmosphere. This is accomplished by evaluating equations 1-12 separately for each grid cell in a distributed isotope enabled global hydrologic simulation, then summing output fluxes globally. Total continental runoff isotope composition is calculated as
4

13 R
Q
=
∑ x , y
R
Q
( x , y ) f
Q
( x , y ) P ( x , y ) A
L
( x , y )
,
∑ x , y f
Q
( x , y ) P ( x , y ) A
L
( x , y ) and the global continental evapotranspiration flux is calculated as
14 R
ET
=
x , y
R
ET
( x , y )(1 − f
Q
( x , y )) P ( x , y ) A
L
( x , y )
,
x , y
(1 − f
Q
( x , y )) P ( x , y ) A
L
( x , y ) where A
L
( x,y ) is the area within each grid cell that is land and fluxes are summed over longitude ( x ), latitude ( y ).
Values obtained from 13 and 14 are compared with values obtained from a mass balance of the oceans ( 23 ). The isotope ratio of ocean evaporation, R
E(O)
, is calculated as in equations 4 and 10, with the isotope composition of sea water assumed equal to zero and C
K taking a range between 2 and 6.5 ( 32 , 33 ). The isotope ratio of global oceanic evaporation is calculated as
15 R
E ( O )
=
∑ x , y
R
E ( O )
( x , y ) E
O
( x , y ) A
O
( x , y )
∑ x , y
E
O
( x , y ) A
O
( x , y )
, where E
O is the yearly bulk evaporation occurring at longitude ( x ) and latitude ( y ), and
A
O
( x,y ) is the area within each grid cell that is ocean . The isotopic ratio of global precipitation over the oceans and land is calculated as
16a R
P ( O )
=
∑ x , y
R
P ( O )
( x , y ) P
O
( x , y ) A
O
( x , y )
∑ x , y
P
O
( x , y ) A
O
( x , y )
,
16b R
P ( L )
=
∑ x , y
R
P ( L )
( x , y ) P
L
( x , y ) A
L
( x , y )
∑ x , y
P
L
( x , y ) A
L
( x , y )
, where P
O
( x,y ) and P
L
( x,y ) are the annual precipitation falling over the oceans and land respectively. Note that in 15 and 16, values of R
E(O)
, R
P(O)
, and R
P(L)
, are flux weighted annually using equations 7 and 8. Finally, a global mass balance of the oceans and atmosphere gives
17 R
Q
=
R
E ( O )
E
O
− R
P ( O )
P
O ,
E
O
− P
O and
5

18 R
ET
=
R
P ( O )
P
O
+ R
P ( L )
P
L
− R
E ( O )
E
O
P
O
+ P
L
− E
O
.
Thus equations 17 and 18 provide estimates of global continental runoff and evapotranspiration flux that are independent of the values calculated overland with equations 13 and 14.
Simulation of HDO pools and fluxes:
Based on the above representation of the HDO cycle over both continents and oceans, a set of isotope enabled global hydrologic simulations are performed to determine the values of two hydrologic unknowns: i ) the fraction of evapotranspiration that passes though the plant stomata ( f
T/ET
), and ii ) the fraction of evaporation that occurs from soils
( f
B/BM
). Within each cell, all inputs are distributed between bound and mobile outputs, i.e
(1f
I
) = ( f
T
+ f
E(B)
) + ( f
E(M) mobile water pools, i.e. f
+f
Q
), and leakage defines the flux of water from the bound to
L
= (1 f
I
) - ( f
T
+ f
E(B)
) = ( f
E(M) evapotranspiration flux that is transpiration as
+f
Q
). We define the fraction of total
19 f
T / ET
=
T
I + T + E
B
+ E
M
= f
I
+ f
T f
T
+ f
E ( B )
+ f
E ( M )
, and the fraction of evaporation that occurs from soils as
20 f
B / ET
=
E
B
E
B
+ E
M
= f
E ( B ) f
E ( B )
+ f
E ( M )
.
Equations 19 and 20 can be rearranged by noting that the total evapotranspiration flux fraction is defined by (1f
Q
), thus f
T
= f
T/ET
(1f
Q
). Similarly, the evaporation fraction
(bound plus mobile) is defined by (1f bound pool is f
E(B)
= f
B/BM
Q
f
I
- f
(1f
Q
f
I
- f
T/ET
(1f
Q
T/ET
(1f
Q
)), and the evaporation from the
)), while evaporation from the mobile pool is f
E(M)
= (1f
B/BM
)(1f
Q
f
I
- f
T/ET
(1f
Q
)). Thus, once equations 19 and 20 are rearranged and inserted into the framework above, the hydrologic partitioning and isotope values for all pools and fluxes are determined based on f
T/ET
, and f
B/BM
, given values of f
Q
and f
I
.
Values of f
T/ET
, and f
B/BM
were estimated though isotope enabled global hydrologic simulation, with the above equations evaluated at 2x2 degree resolution globally.
Uncertainty was assessed though a Monte-Carlo approach, wherein a set of simulations was created, with variability between simulation input parameters determined by observational and theoretical uncertainties (described below). Simulations were generated until 1000 were accurately solved (Figure S1), with the values of f
T/ET
, and f
B/BM found though non-linear least squares minimization of differences between equations 13 and 17 and equations 14 and 18. Simulations were only considered accurately solved if the final values R
ET and R
Q based on continental catchment flux partitioning were less then 0.001
‰ different from the ocean and atmosphere mass balance results and f
T/ET
and f
B/BM
were both between zero and one
For each simulation a single year of bulk global precipitation from the Global
Precipitation Climatology Project (GPCP) ( 28 ) and ocean evaporation from the
Objectively-Analyzed air-sea flux (OAFlux) dataset ( 34 ) was resampled from the period
6

between 2005-2012. Over land, MERRA reanalysis estimates ( 35 ) of evapotranspiration were also resampled from 2005-2012 for each simulation, with continental evapotranspiration scaled linearly to match the ocean evaporation and total global precipitation to ensure mass budget closure. In continental grid cells where yearly evapotranspiration totals exceeded estimated yearly precipitation totals, evapotranspiration was set equal to precipitation. To each resampled set of bulk flux estimates, we add random noise of zero mean with a standard deviation based on the observational uncertainty at that location to each grid cell (observational errors are included with the GPCP, OAFlux and MERRA data files). Gridded runoff values were then calculated for each cell as precipitation minus scaled evapotranspiration values.
This resampling with known observational errors added to each simulation’s bulk fluxes propagates both inter-annual variability in bulk flux amounts and observational uncertainties. In this study we only consider datasets from GPCP, OAFlux, and MERRA
(overland), however the variability that arises from resampling years at random (e.g. oceanic precipitation varies from 372,000 km 3 to 393,000 km 3 ) is larger than reported differences between different data products ( 1 ).
Many different studies have assessed interception, and we approximate this as f
I
≈ 19% of incident precipitation. For each simulation a global value of f
I
was drawn from a normal distribution with mean 0.19 and standard deviations of 0.09, with this range based on the mean and standard deviation of canopy interception studies in ( 27 ). This range of interception values is consistent with values used in previous studies investigation partitioning ( 20 , 6 ). Though intercepted fractions are constant across each simulation, the interception in a grid cell may not exceed evapotranspiration, and in a small number of cells interception was decreased to equal evapotranspiration. Finally, the connectivity between water available for plant use and mobile water is unknown ( 16 , 7 ), and a value of c for equation 5 is randomly drawn between 0 and 1 for each simulation.
Thus spatially varying values of P, ET , f
Q
, and f
I
, are determined for each simulation.
The isotopic value of input precipitation (Fig S2), and its associated uncertainty each month are modeled based on monitoring station data ( 25 ). The D/H ratio of surface vapor (Fig S2B) is estimated from bias-corrected retrievals of the HDO vapor pressure from the Tropospheric Emissions Spectrometer (TES). These bias-corrected values have a mean prediction error of 13.4
‰, with this uncertainty used to create random variability within each simulation ( 23 ). TES retrievals over the period 2005-2012 were averaged to produce a 2x2 degree resolution monthly estimate of mean climatological R
A
.
For each simulation, a kinetic fractionation parameter C
K was drawn from at random from uniform distribution between 2 and 6.5 over the oceans, and a second value was drawn at random between values of 2 and 12.5 overland ( 32 ). A uniform distribution is chosen for the fractionation factors because these have theoretical bounds of 2-6.5 for over open water and up to 12.5 for over land ( 32 ), however between these commonly used bounds we have little reason to justify preference and thus wish to make no value more likely than any other. Temperature and surface normalized relative humidity were estimated based on TES retrievals, with random noise (zero mean, with standard deviations of ±2° for temperature and ±10% for h ) added to each TES retrieval to generate different realizations of surface conditions for each simulation ( 23 ). A Beta distribution is used to
7

add noise to the humidity estimate because it is bound between 0 and 1, while temperature does not have bounds and a normal distribution is used.
Based on this global HDO budget, terrestrial evapotranspiration is determined to be –48.7±1.3‰ (1 σ ) and terrestrial runoff is determined as – 54.7±.3.9 ‰ (isotope values reported in δ notation where δ D= R/R std
-1, with R the D/H isotope ratio and std the
VSMOW standard). Published δ D values of large rivers range from -137‰ to -45‰ (25 th to 75 th percentiles of the 23 largest measured rivers) with a median value -78‰, though this does not include any seasonal or volume weighting ( 23 ). Flux isotope ratio values reported here are better constrained than those reported in ( 23 ) because only simulations involving physically realistic terrestrial flux partitions (e.g. T ≤ ET and E ≤ ET ) are considered here, and this considerably constrains possible continental runoff and evapotranspiration δ D values (Fig. 1).
Model Assumptions and Limitations:
The need to evaluate of our unified isotope hydrology framework on a gridded representation of the terrestrial hydrosphere places specific limitations on our analysis.
Our approach assumes a spatially invariant f
T/ET
ratio because only global, and not local
(grid cell), runoff and evapotranspiration flux isotope ratios are known, however local values are calculated (Fig S2C and Fig S2D). Because local runoff and evapotranspiration bulk fluxes are known, values of T, E
B
, and E
M
, fluxes do vary spatially (Fig S3). This may bias our results due to nonlinearities in equations 6 and 9.
However, no global trend has been observed between f
T/ET
ratios and climate ( 6 ) and the relationship between f
T/ET
and vegetation cover is poorly constrained ( 4 ), thus we do not attempt to prescribe a spatially varying value for this flux ratio here. We also assume a spatially invariant value for f
B/BM
and c for similar reasons. Though these values may also vary spatially, little research has addressed controls on these hydrologic properties over large scales. In the absence of constraints, our global assessment represents an important, if simplified, attempt at hydrologic flux partitioning constrained by available data, and a significant advance over previous attempts that did not include these variables at all.
Because all fluxes into and out of each grid cell are spatiotemporally weighted to obtain global fluxes, our results are most sensitive to those latitudes and regions that contribute most to the global hydrologic cycle (e.g., tropical and mid-latitude forests), suggesting that improved understanding of spatial controls on hydrologic partitioning within these areas will be most valuable in refining the estimates derived here. Our approach also does not allow for vegetation to utilize mobile waters, though this has been observed in riparian vegetation. Additionally, we consider interception to be equal to evaporation, which assumes that all intercepted precipitation evaporates. It is possible that partially evaporated, and therefore partially fractionated, intercepted water may drip or be shaken by wind to the ground. This effect is likely small, and in our approach we remain consistent with previous studies ( 3 ) and do not consider it.
Though only mean annual bound and mobile water pool isotope values can be determined using our annual isotope mass balance framework, fluxes are resolved at the monthly scale and assumed constant over shorter timescales. This takes into account seasonal (but not daily) variation in climate and local D/H isotope ratios. Because of short-term (hourly or daily) changes in meteorological conditions, soil conditions, and
8

plant stress, non-steady state conditions likely occur and affect isotopic fractionation during evaporation from plant leaves ( 36 ). Similarly, short-term variation in surface conditions will also influence both soil and surface water evaporation isotopic composition. Lack of consideration of these short-term dynamics may also bias our results, particularly those values of R
T
, R
E(M)
, and R
E(B)
.
However seasonal variations in forcing conditions are expected to be much wider then daily variations, and monthly representation mean meteorological values likely captures the scale of environmental variability required to estimate climatological mean fluxes.
Because the continental water pools in our simulations are considered isotopically well-mixed across the annual cycle, seasonal and shorter-term variation in runoff fractions ( Q/P ) may bias our runoff isotope ratios where this variation is coupled with systematic shifts in precipitation isotope ratios. This effect has been identified, for example, in catchments with high-elevation headwaters throughout the United States, where runoff consists primarily of winter time precipitation with low R
P values ( 37 ).
Because the mass balance framework on which our analysis is based is not valid at subannual timescales we cannot account for this possible effect in the current analysis.
However because the majority of the globe’s hydrologic cycle occurs within the tropics this bias is likely small. Additionally, though we follow previous studies assumption that transpiration is not fractionated, large-scale annual observations of this phenomena are lacking. In our framework, we estimate the contribution of non-fractionating and fractionating fluxes, and assume that the non-fractionating fluxes approximate transpiration amounts. Finally, because our calculations do not consider the downstream accumulation of surface water (i.e. runoff is directly routed to the ocean), cells receiving large amounts of upstream runoff relative to local inputs are likely to have slightly biased
R
M
. This may have a small effect on the calculated surface water evaporation isotopic composition, however because of the small fraction of the global surface area covered by such systems we expect this effect is negligible in the global mass balance.
Though our approach requires simplifying assumptions about spatially and temporally varying hydrologic process, we believe it represents a judicious tradeoff between constraints provided by currently available data and opportunities to evaluate critical hydrologic unknowns. We are able to represent all inputs and outputs of the entire terrestrial hydrosphere, and thus incorporate all watersheds worldwide, even those with no measurements of D/H isotope ratios. Furthermore, our approach requires no assumptions about plant water use strategies (beyond specifying that transpiration is extracted from the bound pool only) and stomatal function, which are poorly constrained parameterizations needed in global Earth system modeling approaches. By evaluating our framework though a set of Monte-Carlo simulations that include consideration of potential errors in all input parameters, our results include bounded uncertainty envelopes. Though these envelopes are wide, they provide useful constraints for global
Earth system models by elimination possible hydrologic parameters space and may be further refined in future work. Finally, we note that we make no formal statistical inference, and the Monte-Carlo simulations we undertake do not account for possible uncertainty in the physical relationships that we have chosen, thus our results should be considered within the context of the assumptions outlined here and inherent in the massbalance framework described above.
9

Fig. S1.
Convergence of global simulations: 1000 simulations of global the HDO cycle, the isotope ratio of ( A ) continental evapotranspiration and ( B ) continental runoff composition. Mean values across all runs are shown in yellow with estimated standard deviations as dashed lines.
10

Fig. S2
Isotope ratio of land fluxes: ( A ) Flux weighted mean precipitation D/H ratios ( 25 ). ( B )
Bias corrected estimates of surface vapor D/H isotope ratios ( 23 ). ( C ) Calculated difference between local runoff composition and local input precipitation. ( D ) Calculated difference between local evaporation composition and local input precipitation.
11

Fig. S3
Partitioned estimates of global hydrologic fluxes : Global mean annual precipitation
( A ) is partitioned into either runoff ( B ) or evapotranspiration sub-components of interception ( C ), transpiration ( D ), evaporation from soils ( E ), and evaporation from surface waters ( F ).
12

References and Notes
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44 , 23–49
(2008).
33. L. Merlivat, J. Jouzel, Global climatic interpretation of the deuterium-oxygen 18 relationship for precipitation. J. Geophys. Res.
84 , 5029–5033 (1979).
34. L. Yu, R. A. Weller, Objectively Analyzed Air–Sea Heat Fluxes for the Global
Ice-Free Oceans (1981–2005). Bull. Am. Meteorol. Soc.
88 , 527–539 (2007).
35. M. M. Rienecker et al.
, MERRA: NASA’s Modern-Era Retrospective Analysis for
Research and Applications. J. Clim.
24 , 3624–3648 (2011).
36. G. D. Farquhar, L. A. Cernusak, On the isotopic composition of leaf water in the non-steady state. Funct. Plant Biol.
32 , 293–303 (2005).
37. G. J. Bowen, C. D. Kennedy, Z. Liu, J. Stalker, Water balance model for mean annual hydrogen and oxygen isotope distributions in surface waters of the contiguous United States. J. Geophys. Res.
116 , 1–14 (2011).
13