GEOPHYSICAL RESEARCH LETTERS, VOL. 38, L14603, doi:10.1029/2011GL047652, 2011
Exact evaluation of gross photosynthetic production
from the oxygen triple‐isotope composition of O2:
Implications for the net‐to‐gross primary production ratios
Maria G. Prokopenko,1 Olivier M. Pauluis,2 Julie Granger,3 and Laurence Y. Yeung1,4
Received 15 April 2011; revised 26 May 2011; accepted 26 May 2011; published 19 July 2011.
[1] The oxygen triple‐isotope composition of dissolved O2
provides an integrative method to estimate the rates of Gross
Photosynthetic Production (GPP) in the upper ocean, and
combined with estimates of Net Community Production
(NCP) yields an estimate of the net‐to‐gross (NCP/GPP)
production ratios. However, derivations of GPP from
oxygen triple‐isotope measurements have involved some
mathematical approximations. We derive an exact
expression for calculating GPP, and show that small errors
associated with approximations result in a relative error of
up to ∼38% in GPP, and up to ∼50% in N/G. In open
ocean regimes with low primary production, the observed
magnitude of the error is comparable to the combined
methodological uncertainties. In highly productive
ecosystems, the error arising from approximations becomes
significant. Using data collected on the Bering Sea shelf,
we illustrate the differences in GPP estimates in both high
and low productivity regimes that arise from exact and
approximated formulations. Citation: Prokopenko, M. G.,
O. M. Pauluis, J. Granger, and L. Y. Yeung (2011), Exact evaluation
of gross photosynthetic production from the oxygen triple‐isotope
composition of O2: Implications for the net‐to‐gross primary production ratios, Geophys. Res. Lett., 38, L14603, doi:10.1029/
2011GL047652.
1. Introduction
[2] Accurate estimates of biological Primary Production
(PP) are essential for understanding the effects of climatic
and environmental forcings on terrestrial and aquatic ecosystems. Estimating the primary production rates in the
ocean has been notoriously difficult [e.g., Laws et al., 2000;
Robinson et al., 2009; Quay et al., 2010], and often multiple
approaches are employed, broadly classified into 3 categories: shipboard incubations, satellite observations of ocean
color, and in situ geochemical mass balance methods. For
detailed descriptions of these methods, the reader is referred
to review papers [Bender et al., 1987; Laws et al., 2000; Luz
1
Department of Earth Sciences, University of Southern California,
Los Angeles, California, USA.
2
Courant Institute of Mathematical Sciences, New York University,
New York, New York, USA.
3
Geosciences Department, Princeton University, Princeton, New
Jersey, USA.
4
Now at Department of Earth and Space Sciences, University of
California, Los Angeles, California, USA.
Copyright 2011 by the American Geophysical Union.
0094‐8276/11/2011GL047652
and Barkan, 2009; Robinson et al., 2009; Quay et al., 2010,
and references therein].
[3] Gross primary production (GPP) is a measure of the
amount of energy derived from the oxidation of water
through the catalytic water‐splitting function of photosystem II. Net community production (NCP) is defined as the
amount of chemical energy produced by primary producers
minus that respired by authotrophs and heterotrophs; thus,
it represents the biological organic carbon input to the
ocean. The ratio of net to gross primary production (NCP/
GPP,), to first order, reflects the organic carbon export efficiency out of the ocean surface. Rates of NCP can be determined by measuring the biological supersaturation of O2
[Craig and Hayward, 1987; Emerson, 1987; Spitzer and
Jenkins, 1989; Emerson et al., 1993], whereas measurements
of the three stable isotopes of oxygen (via 17O/16O and 18O/16O
ratios, X17 and X18) yield estimates of GPP [Luz and Barkan,
2000, 2005, 2009] when the rate of gas exchange is known.
[4] While a powerful integrative approach, derivation of
GPP rates with the oxygen triple isotope composition constitutes an “ill‐posed” problem in a mathematical sense, as it
is highly sensitive to small errors, such as those introduced
by commonly used approximations. We derive the exact
expression relating the oxygen triple‐isotope composition to
GPP and show that depending on the approximation
applied, previous formulations lead to an error up to 38% in
the estimates of GPP, causing a corresponding overestimation
in N/G ratios of up to 50%. In the typical oligotrophic open
ocean environment, the approximation error is comparable to
the combined analytical and gas exchange uncertainties. In
ecosystems with elevated export ratios GPP underestimation
results in derived NCP/GPP values higher than physiologically plausible [Bender et al., 1999; Laws et al., 2000; Halsey
et al., 2010].
2. Basis of the Triple‐Isotope Method
[5] The oxygen triple‐isotope method is based on difference in 17O/16O and 18O/16O abundances between photosynthetic O2 and atmospheric O2 [Luz et al., 1999; Luz and
Barkan, 2000]. For a detailed description of the method the
reader is referred to recent papers of [Luz and Barkan, 2005,
2009]. Briefly, photosynthesis produces O2 with 17O and
18
O abundances similar to those of source water [Guy et al.,
1993; Helman et al., 2005; Eisenstadt et al., 2010]. Oxygen
consumption (mostly through respiration) enriches O2 18O
(the Dole effect [Dole, 1935; Dole et al., 1954]), accounting
for d18O of atmospheric O 2 of 23.5 ‰ [Kroopnick and
Craig, 1972]. Barkan and Luz [2005] recently revised
atmospheric d18O to 23.88 ‰ and reported the d 17O of
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12.08 ‰ relative to Vienna Standard Mean Ocean Water
(VSMOW). The d18O and d17O are defined either as d*O =
( X*air* − 1)*1000, where X* is ratio of 17/18O/16O or as d*O =
XVSMOW
*
(XVSMOW
Xair
* − 1)*1000. We adopt here the latter definition
[Barkan and Luz, 2003]. Atmospheric O2 also bears a separate mass‐independent signature that arises from the photochemistry of O3 and CO2 in the stratosphere [Thiemens and
Heidenreich, 1983; Yung et al., 1991; Thiemens et al.,
1995], and is the basis of the oxygen triple‐isotope
method [Luz et al., 1999; Luz and Barkan, 2000]. Tropospheric O2 is mass‐independently depleted in X 17 relative
to X 18, in contrast to nascent O2 produced by photosynthesis
[Luz et al., 1999; Luz and Barkan, 2000; Helman et al., 2005;
Luz and Barkan, 2005, 2009; Eisenstadt et al., 2010]. If the
rate of gas exchange between the surface ocean mixed layer
and the atmosphere is known, rates of GPP can be estimated
from measurements of oxygen triple‐isotope distributions in
surface waters [Luz and Barkan, 2000, 2005, 2009].
[6] The magnitude of the 17O‐to‐18O enrichment has been
historically defined in two ways, either through the difference in isotopic d‐values (1):
"
D17 O ! 17
¼ ! O " "′*!18 O
103
ð1Þ
or through a difference in the logarithms of isotopic ratios
[Miller, 2002; Luz and Barkan, 2005] (2):
# 17 $
# 18
$
# 17
$
! O
X
D
! O
þ 1 ¼ ln 17
¼ ln
þ 1 " "* ln
103
Xstd
106
103
# 18 $
X
" "* ln 18
Xstd
17
ð2Þ
Here, 17D (or D17O), reported in per meg, is the magnitude
of deviation from the mass‐dependent relationship between
17
O and 18O [Meijer and Li, 1998; Miller, 2002]; l = 0.518
is a constant for the mass‐dependent fractionation between
17
O and 18O during respiration [Luz and Barkan, 2005,
2009]; l′ is a quantity similar to l, but depends on the
isotope ratios of the reference material [Miller, 2002; Angert
et al., 2003].
[7] The expression for estimating rates of GPP based on
the triple‐isotope composition of O2, originally proposed by
Luz and Barkan [2000] for the D17O definition of 17O
excess has been applied in subsequent studies for the 17D
definition [e.g., Hendricks et al., 2004; Luz and Barkan,
2009]:
17
Ddis " 17Deq
G
& 17
kO2eq
Dp " 17Ddis
excess (equations (1) and (2)) differs from the exact solution
of the simple box model originally envisioned by Luz and
Barkan [2000].
3. Exact Derivation of GPP From Oxygen Triple‐
Isotope Measurements
[8] We derive an exact expression of GPP from the three
O isotope concentrations, considering an oceanic mixed
layer of constant thickness h. The rate of change of the
dissolved oxygen, [O2], is given by:
h
%
&
@ ð½ O 2 ( Þ
¼ G " R " k ½O2 ( " ½O2 (eq
@t
ð4Þ
where G is the GPP flux and R is the respiration flux. Expressing isotope ratios as X * = *O16O/16O16O, where the
asterisk refers to a specific isotope, 17O or 16O, the mass
balance for either isotope can be written as:
h
!
"
!
"
*
@ OXdis
* " k Odis Xdis
* " Oeq Xeq*
¼ GXp* " R#*Xdis
@t
ð5Þ
where O refers to [O2]. Photosynthetic production for *O is
given by GX*p, where X*p is the isotope ratio of photosynthetically produced O2 [Luz and Barkan, 2000, 2005;
Eisenstadt et al., 2010]. Respiration of *O is expressed as
R* = Ra*, where a* is the respiration fractionation factor
for either isotopologue. In the auxiliary material, we show
that equations (4) and (5) can be combined to obtain the rate
of change of 17D:1
"
#
17
18
Xp17 " Xdis
Xp18 " Xdis
@ 17 D
¼G)
hOdis
""
17
18
@t
Xdis
Xdis
"
#
17
17
18
18
Xeq
" Xdis
Xeq
" Xdis
"
"
" kOeq )
17
18
Xdis
Xdis
ð6Þ
[9] Expression (6) yields the rate of change of 17D in
the mixed layer that results from primary production and
gas exchange, and is independent of respiration (auxiliary
material). The quantity l (same as in equation (2)) is
#17
expressed as l = 11 "
" #18 . In the absence of both primary
production and gas exchange (i.e., G = 0 and k = 0) the 17D
is conserved [Miller, 2002, Angert et al., 2003]. In the
surface ocean, (G ≠ 0, k ≠ 0) 17D is not conserved because
of the injection of oxygen masses from photosynthesis and
gas exchange with atmospheric O2. Setting the left‐hand
side of equation (6) to zero, we obtain a steady state
18
expression relating GPP to values of X17
dis and X dis, which are
measured directly (7):
ð3Þ
where G/kOeq is the gross O2 production flux (GPP) divided by
the atmospheric O2 invasion, k is the piston velocity for O2,
O2eq is [O2] at equilibrium with atmosphere (at a given temperature, pressure, and salinity), and subscripts dis, eq and p stand
for dissolved, equilibrium and photosynthetic O2, respectively.
As we show below, equation (3) using either definition of 17O
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G
¼
kOeq
17
17
Xdis
" Xeq
17
Xdis
17
Xp17 " Xdis
17
Xdis
""
""
18
18
Xdis
" Xeq
18
Xdis
18
Xp18 " Xdis
ð7Þ
18
Xdis
1
Auxiliary materials are available in the HTML. doi:10.1029/
2011GL047652.
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Table 1a. Input Parameters
17
Oeq@ 10°Ca
d*O, ‰
18
Oeq@ 10°Cb
0.411
17
O excess, per meg
Slope in calculations of
17
OVSMOWc
−11.931
(−11.881)
173
0.516
0.779
8
0.518
O excess
17
18
OVSMOWc
−23.320
O Photosynthesisd
17
−11.902
(−11.858)
249
0.518
(0.516)e
a
Luz and Barkan [2009].
Benson and Krause [1984].
Barkan and Luz [2005].
d
Calculated according to definition in equation (2).
e
Luz and Barkan [2005].
b
c
or in d notation:
#
$
#
$
10"3 !17 Oeq þ 1
10"3 !18 Oeq þ 1
" " 1 " "3 18
1 " "3 17
G
10 ! Odis þ 1
10 ! Odis þ 1
$
# "3 18
$
¼ # "3 17
kOeq
10 ! Op þ 1
10 ! Op þ 1
"1 ""
"1
10"3 !17 Odis þ 1
10"3 !18 Odis þ 1
Equation (7) is the exact expressions for the GPP rate in a
steady‐state system.
4. Comparison of Exact and Approximated
Formulations
[10] Equation (7) can be expressed as:
G
F1 " "F2
¼
kOeq F3 " "F4
ð8Þ
where F1‐4 is a function of isotopic ratios. To facilitate comparison, we re‐write expression (3) for the D17O (equation (9a))
and 17D (equation (9b)) notations in a similar form:
G
¼
kOeq
17
17
Xdis
" Xeq
17
Xstd
!
17
Xp17 " Xdis
17
Xstd
!
X 17
ln dis
17
Xeq
!
""
"
′
"′
18
18
Xdis
" Xeq
18
Xstd
18
Xp18 " Xdis
18
Xstd
!
!
ð9aÞ
and
G
¼
kOeq
ln
Xp17
17
Xdis
!
X 18
" " ln dis
18
Xeq
" " ln
Xp18
18
Xdis
!
!
ð9bÞ
Comparing to the exact expression for G/kOeq given in
equation (7), we note that F(1‐4) in equations (9a) or (9b) is
% &
*"X *
*
*"X *
*
approximated as either XA B ≈ XA "XB or XA B ≈ ln XA ,
Xstd
*
XB*
XB*
XB*
where XA is either X *eq or X *p and X*B = X*
dis. Note that the
value of X d*is reflects both 1) two‐endmember mixing
between atmospheric O2 (X*
eq) and photosynthetic O2 (X*
p)
and 2) mass‐dependent kinetic fractionation by respiration.
These processes are represented in equation (7), but not in
either equations (9a) or (9b). For instance, the numerator and
denominator of equation (9b) represent mass fractionation
laws relating O2dis to O2eq and O2p, respectively, but they do
not account for two‐endmember mixing well [Miller, 2002].
[11] To illustrate the effect of the approximations on the
value of G/kOeq we analyze an individual data point, with
a measured d18 Odis of −4.899 ‰ and d 17Odis of −2.439 ‰.
In this comparison, we assumed l′ = l = 0.518 (for a detailed
discussion of the difference between the two quantities, see
Miller [2002], Angert et al. [2003], and Kaiser [2011]). The
constants used in the example given below are listed in
Table 1a. Taken individually, the approximations for the
F(1‐4) in equation (7) vs. equations (9a) and (9b) are
accurate (Table 1b). However, the expression for GPP involves
the differences F1 − lF2 and F3 − lF4. Comparing F(1‐4) from
equations (7) and (9a) and (9b), we find the relative error
between exact and the approximated versions for the term
F1 − lF2 in the numerator is fairly small, ∼5 to ∼8%, for
linear and log approximations, respectively. Conversely,
the differences F3 − lF4 in (9a) and (9b) are ∼23% smaller
and ∼43% larger, respectively, than the equivalent terms
in equation (7). This leads to an underestimation of the
G/kOeq by >33% in the log approximation and overestimation by 20% in the linear approximation (Table 1b).
[12] The magnitude of the relative error introduced by the
approximated equations (9a) and (9b) was computed for a
range of G/kOeq and N/G (Figure 1). The linear notation,
D17O (equation (9a) [Luz and Barkan, 2000] results in a
significantly smaller error in the estimates of the G/kOeq
Table 1b. Comparison of Terms F1‐4 and Calculations Based on Exact and Approximated Equationsa
Terms F1–F4
(see text for explanation)
Exact
(Equation (7))
Linear Approximation
(Equation (9a))
% Error Linear
Approximation
Log Approximation
(Equation (9b))
% Error Log
Approximation
F1
F2
(F1‐l*F4)
F3
F4
(F3‐l*F4)
GPP/kOsat
NCP/GPP (for DO2superSat = 0.44)
−0.0028685
−0.0057275
0.0000984
−0.0094864
−0.0185112
0.0001024
0.96
0.46
−0.0028614
−0.0056994
0.0000908
−0.0094633
−0.0184205
0.0000786
1.16
0.38
−0.24%
−0.49%
−7.66%
−0.24%
−0.49%
−23.28%
20.40%
−16.94%
−0.0028644
−0.0057112
0.0000940
−0.0095317
−0.0186847
0.0001470
0.64
0.69
−0.14%
−0.29%
−4.43%
0.48%
0.94%
43.53%
−33.39%
50.13%
a
See text for definitions of terms.
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Figure 1. Relative error (RE) as a function of GPP/k*Osat and the NCP/GPP ratios. (a) Linear approximation; (b) logarithmic approximation, d17Op = −11.902 ‰, l = 0.518 [Barkan and Luz, 2005]; and (c) logarithmic approximation, d17Op =
−11.858 ‰, l = 0.516.
than its logarithmic counterpart, 17D (equation (9b)). For
typical open ocean conditions, the D17O formulation only
deviates by <5% from the exact formulation (though the
magnitude of the deviation depends on a value of l′ [Kaiser,
2011]). We also note here that the magnitude of the relative
error in log approximation depends on the actual value
of constants (such as l, relating d*Odis, d*Op and d*Oeq
to each other). If the oxygen triple‐isotope composition of
photosynthetic O2 is related to that of atmospheric O2
through the slope of 0.516 (instead of 0.518, Table 1a [Luz
and Barkan, 2005]), the discrepancy between equations (7)
and (9b) is reduced to ≤∼15% (compare Figures 1b and 1c).
Further precise measurements of the fundamental constants
are needed to fully constrain the oxygen triple‐isotope
composition.
[13] The exact and approximated expressions for G/kOeq
were applied to data collected in spring 2007 within the
mixed layer on the Bering Sea shelf (Figures 2a–2c) before
and during the spring bloom. The NCP/GPP ratios were
computed with NCP obtained as shown in equation (10),
which is derived from equation (4) with a steady state
assumption:
G " R ¼ NCP ¼ "kOeq * DObio :
tive ecosystems, errors prove more significant, such that the
exact expression should be used to estimate PP and export
efficiency.
5. Conclusions
[15] Here we proposed an exact expression for quantifying the rates of GPP, which is free of inaccuracies introduced
by previously used approximations. We applied the newly
derived expression to a data set collected during spring
blooms on the Bering Sea shelf, and demonstrated that
application of the exact expression is necessary to obtain
ð10Þ
where DObio is biological supersaturation of O2 in the
mixed layer. The DObio derives from concurrent measurements of O2/Ar gas ratios [Craig and Hayward, 1987; see
Luz et al., 2002; Hendricks et al., 2004, 2005]. NCP/GPP
ratios obtained with logarithmic approximation (Figure 2c)
overestimate oxygen NCP/GPP ratios by ∼50%, leading
to erroneously high ratios that exceed biologically feasible
values of ∼0.4 [Halsey et al., 2010] to ∼0.5 [Bender et al.,
1999].
[14] Published studies to date have applied the O triple‐
isotope method mostly in the open ocean‐lower productivity ecosystems with low export efficiencies. Applying
equation (9a), the linear version of equation (3), led to a
modest error of <5%. The error arising from using the
logarithmic form (equation (9b)) in oligotrophic ecosystems,
while more sizable, was comparable in magnitude to the
typically reported uncertainty of ∼30%, associated with
parameterization of air‐sea gas exchange and analytical precision [Hendricks et al., 2004, 2005]. However, when the
oxygen triple‐isotope method is applied in highly produc-
Figure 2. (a) Relative error (RE) introduced in GPP rates
estimates by linear and log approximations from data collected in the spring mixed layer on the Bering Sea shelf.
(b) Comparison of GPP/k*Osat calculated using exact and
approximated equations. (c) NCP/GPP ratios calculated
using exact expression (equation (7)) and two different
approximated equations (9a) and (9b). The data point which
is used in calculations in Tables 1a–1c is circled. The dashed
line represents the maximum physiologically feasible NCP/
GPP value.
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PROKOPENKO ET AL.: EVALUATION OF GPP FROM O2 ISOTOPES
more accurate net‐to‐gross production ratios and carbon
export efficiency in highly productive ecosystems.
[16] Acknowledgments. We gratefully acknowledge generous help
of Michael Bender and his lab at the Department of Geosciences, Princeton
University, with running oxygen triple‐isotope and O2/Ar analyses presented in this paper. This work would not be possible without much appreciated technical assistance of Bruce Barnett. We also wish to thank Eugeni
Barkan, Boaz Luz, David (Roo) Nicholson and two anonymous reviewers
for helpful discussion and insightful comments on the earlier versions of
this manuscript. MGP was supported by NSF grants OCE‐ 0961207
awarded to W. Berelson and OPP‐0612198 awarded to D. M. Sigman.
JG was supported by an NSF grant OPP‐0612198 awarded to D. M. Sigman.
LYY was supported by NSF grant OCE‐0934095 to W. Berelson.
[17] The Editor thanks two anonymous reviewers for their assistance in
evaluating this paper.
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J. Granger, Geosciences Department, Princeton University, Princeton, NJ
08544, USA.
O. M. Pauluis, Courant Institute of Mathematical Sciences, New York
University, 251 Mercer St., New York, NY 10012, USA.
M. G. Prokopenko, Department of Earth Sciences, University of
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L. Y. Yeung, Department of Earth and Space Sciences, University of
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