## 1. Introduction to a New Three-End-Member Mixing Model

[2] *Redfield* [1958] proposed an idealized molar formula for an average marine plankton, i.e., (CH_{2}O)_{106}(NH_{3})_{16}(H_{3}PO_{4}). Oxidation or remineralization of one mole of this idealized plankton can be represented by the following overall net reaction: (CH_{2}O)_{106}(NH_{3})_{16}(H_{3}PO_{4}) + 138 O_{2} → 106 CO_{2} + 16 NO_{3}^{−} + H_{2}PO_{4}^{−} + 17 H^{+} + 122 H_{2}O. Molar ratios among remineralized P, N, C_{org}, and consumed O_{2} (P/N/C_{org}/−O_{2}) are called remineralization ratios or Redfield ratios. Remineralization ratios of P/N/C_{org}/−O_{2} = 1/16/106/138 for the idealized marine plankton are known as the “traditional Redfield ratios”. Remineralization ratios are useful in explaining the coupled nature of nutrients and carbon cycles in the oceans. They are also important constants needed to estimate conservative tracers such as “NO” and “PO” [*Broecker*, 1974; *Broecker et al.*, 1985]; the preformed nitrate, phosphate, and total dissolved inorganic carbon (NO_{3}^{0}, PO_{4}^{0} and DIC^{0}) [*Chen and Pytkowicz*, 1979; *Li et al.*, 2000]; and the extent of denitrification and nitrogen fixation [*Gruber and Sarmiento*, 1997; *Deutsch et al.*, 2001] in the oceans. It has been assumed that remineralization ratios remain constant over the global oceans. The availability of high quality data of nutrients and carbon chemistry as a result of WOCE/JGOFS global ocean survey provides an opportunity to reevaluate the remineralization ratios of organic matter and challenge the validity of their uniformity in the world oceans. Possible spatial and temporal changes of remineralization ratios in the oceans are the subject of the present study.

[3] *Li et al.* [2000] presented a new two-end-member mixing model to estimate remineralization ratios P:N:C_{org}:−O_{2} = 1:(r_{p}/r_{n}):(r_{p}/r_{c}):r_{p}. Here, r_{p}, r_{n}, and r_{c} are respectively the molar ratios of consumed O_{2} to remineralized P, N, and C_{org} during oxidation of organic matter in the water column, i.e. r_{p} = −O_{2}/P, r_{n} = −O_{2}/N, and r_{c} = −O_{2}/C_{org}. The model essentially involves a multiple-variable linear regression (hereafter simplified as multiple regression) of hydrographic data of water samples within the linear segment of a chosen θ-S (potential temperature versus salinity) plot to obtain r_{p}, r_{n}, and r_{c} values. Hydrographic variables include potential temperature (θ), dissolved oxygen (O_{2}) along with dissolved inorganic phosphate (PO_{4}) or dissolved nitrate (= NO_{3}^{−} + NO_{2}^{−}), or DA (= DIC-Alkalinity/2). The advantage of this model over earlier ones [*Takahashi et al.*, 1985; *Broecker et al.*, 1985; *Minster and Boulahdid*, 1987; and *Boulahdid and Minster*, 1989] is that it requires no knowledge of the end-member values for each variable. As noted by *Minster and Boulahdid* [1987], the way end-member values were chosen in earlier models could be problematic. The model by *Anderson and Sarmiento* [1994] also requires no knowledge of end-member values. However, their assumption that ΔC_{inorg} (carbon increment from dissolution of carbonates) divided by ΔP (phosphate increment from oxidation of organic material) is constant along any neutral surface in their model is unrealistic, especially in water masses above the carbonate saturation depth. As summarized by *Morse and Mackenzie* [1990], the calcite saturation depth is about 4000 ± 500 m in the Indian and Pacific Oceans and 4500 ± 500 m in the Atlantic Ocean. In a saturated or oversaturated water mass, the formation processes of carbonates are completely decoupled from the oxidation process of organic matter. Only in some microenvironments may the oxidation of organic matter enhance the dissolution of carbonates [*Troy et al.*, 1997].

[4] *Hupe and Karstensen* [2000] introduced a linear inverse mixing model with multiple end-members to estimate remineralization ratios in the Arabian Sea. However, the model requires knowledge of end-member values for each variable. The choice of proper end-member characteristics is always subjective, especially for nonconservative tracers. *Shaffer* [1996] fitted global averaged profiles of phosphate, nitrate, and dissolved oxygen to his high-latitude-exchange/interior-diffusion-advection model to estimate remineralization ratios, and he concluded that remineralization ratios change with depth. Using a new diffusion-advection model with source or sink terms and allowing local exchange across neutral surfaces, *Shaffer et al.* [1999] obtained a similar conclusion but the remineralization ratios reach constant values below 1500 m. The model by *Shaffer et al.* [1999] also requires a prior knowledge of end-member characteristics. In contrast, *Broecker et al.* [1985], *Peng and Broecker* [1987], and *Anderson and Sarmiento* [1994] concluded that remineralization ratios do not change much with depth below 400 m.

[5] To use available data more effectively and to cover a greater portion of ocean regions for the evaluation of remineralization ratios, a logical extension of the two-end-member mixing model of *Li et al.* [2000] is to introduce the additional conservative tracer salinity (S) and construct a three-end-member mixing model. In this model, selected data need not be confined in the linear segment of the θ-S diagram as required for any two-end-member mixing model. The model also requires no prior knowledge on the end-member values for each variable.

[6] The following mass balance equations for three conservative tracers, θ, S, and (NO), hold in a water sample that resulted from three-end-member mixing:

Here θ, S, O_{2}, and NO_{3} are observed values for the water sample; subscripts 1, 2, and 3 represent the three end-members; f_{i} is the fraction of the end-member i (= 1, 2, and 3) in the sample; (NO)_{i} is the concentration of conservative tracer “NO” (= O_{2} + r_{n} · NO_{3}; here r_{n} is assumed constant [*Broecker*, 1974]) for the end-member i. First, we solve for f_{1}, f_{2}, and f_{3} from equations (1a) to (1c) in terms of observed θ and S, and end-member values θ_{i} and S_{i}; then substitute the solutions into equation (1d). The result is a linear equation:

Note that the coefficients α_{0}, α_{1}, α_{2}, and D are all functions of θ_{i}, S_{i}, and (NO)_{i} (i = 1, 2, and 3) for the three end-members, and are unknown but stay constant during the mixing process. One may regard end-member values θ_{i}, S_{i}, and (NO)_{i} as mathematical dummies, and they need not have explicit values.

[7] By multiple regression of O_{2}, θ, S and NO_{3} concentration data from a properly selected hydrographic transect that contains three end-members (see the next section), one can estimate the regression coefficients α_{0}, α_{1}, α_{2}, and r_{n} in equation (2a). However, our focus is only on the value of r_{n}. As discussed by *Li et al.* [2000], one can rewrite equation (2a) to make NO_{3} a dependent variable, i.e.,

[8] By multiple regression, one can also obtain independently 1/r_{n} and thus its inverse, designated here as r_{n}′. The r_{n} and r_{n}′ values are always not equal, unless the square of the multiple correlation (R^{2}) is one. The results given in this paper are all averaged values, i.e., (r_{n} + r_{n}′)/2 with the estimated uncertainty of ±[(r_{n} − r_{n}′)/2] or ±[(Δr_{n})^{2} + (Δr_{n}′)^{2}]^{1/2}/2, whichever is larger. Here, Δr_{n} and Δr_{n}′ are the standard errors for r_{n} and r_{n}′, given by any multiple regression program for personal computer (in our case, we used the Statistical Package for the Social Sciences, SPSS). When the square of the multiple correlation (R^{2}) is 0.98 or higher for the multiple regression, the r_{n} and r_{n}′ values always overlap each other within the estimated standard errors.

[9] Similarly, replacing “NO” in equation (1d) with another conservative tracer “PO” (= O_{2} + r_{p} · PO_{4} [*Broecker*, 1974]), and NO_{3} with PO_{4}, one obtains the following:

where A's are similar in formula to α's given beneath equation (2a), except replacing end-member values (NO)_{i} with (PO)_{i}. Again, by multiple regression of O_{2}, θ, S, and PO_{4} data, one can obtain r_{p}, r_{p}′ and (r_{p} + r_{p}′)/2.

[10] When O_{2} data are missing, the r_{p}/r_{n} (= N/P) ratio still can be estimated by multiple regression of NO_{3}, θ, S, and PO_{4} data according to the following equation, which is obtained by eliminating O_{2} from equations (2a) to (2c):

[11] As shown by *Li et al.* [2000], the following relationships are valid for total dissolved inorganic carbon (DIC) and total alkalinity (Alk) in a water sample:

where x and y are respectively cumulative increments of DIC from dissolution of carbonate minerals and from oxidation of organic matter during the transport of a water sample from its source region to the current position; “a” = H^{+}/C_{org} (molar ratio of the produced H^{+} to the remineralized C_{org} during oxidation of organic matter) = (N + P)/C_{org} = r_{c}/r_{n} + r_{c}/r_{p} ≈ r_{c}/r_{n} (since r_{c}/r_{n} ≫ r_{c}/r_{p}); DIC^{0}, Alk^{0}, and O_{2}^{0} are the initial values at the source region. If O_{2}^{0} is the saturated oxygen concentration at the given S and θ of the water sample in equilibrium with air, then (O_{2}^{0} − O_{2}) by definition becomes AOU (apparent oxygen utilization), and DIC^{0} and Alk^{0} are the preformed DIC and alkalinity.

[12] By assuming O_{2}^{0} to be equal to the saturated oxygen in equilibrium with air, and eliminating x and y from equations (3a) to (3c), one obtains the following:

where DA = DIC – Alk/2, and DA^{0} (= DIC^{0} − Alk^{0}/2) is the preformed DA.

[13] The new variable DA (combining capital letters of DIC and Alk) can be visualized as alkalinity-corrected DIC [*Li et al.*, 2000]. Like any other preformed parameter, DA^{0} is also a conservative tracer that changes in water samples only through water mass mixing before the industrial time. Strictly speaking, DA^{0} is no longer a conservative tracer for the shallow part of the ocean where DIC^{0} changes with time due to anthropogenic CO_{2} inputs. However, the increment rate of DIC in the surface mixed layer is only about 0.05% per year [*Winn et al.*, 1998]. Furthermore, the regionwide vertical mixing coefficient for the oceanic thermocline below the surface mixed layer is also finite (0.5 to 2.8 cm^{2}/s, which corresponds to 56 to 94 m of average mixing depth per year [*Li et al.*, 1984]). Therefore, DA^{0} still can be considered as near constant in the lower thermocline waters during a short period of one to five years, and it contains both the preindustrial and cumulative anthropogenic CO_{2} components. DA^{0} for any deep-water sample, where anthropogenic CO_{2} input is negligibly small [*Sarmiento et al.*, 1992; *Wanninkhof et al.*, 1999], can be expressed by

if the water sample is formed by three-end-member mixing.

[14] By substituting equations (1a), (1b), (1c) and (4b) into equation (4a), one obtains

where β_{0}, β_{1}, and β_{2} are similar in formula to α_{0}, α_{1}, and α_{2} in equation (2a), except (NO)_{i} are replaced by (DA^{0})_{i}; and β_{3} = (a/2 + 1)/r_{c}, thus r_{c} = 1/[β_{3} − 0.5(1/r_{n} + 1/r_{p})] ≈ 1/[β_{3} − 0.5/r_{n}]. As will be shown in Table 1, the range of r_{n} is between 8 and 13, therefore the change in r_{n} will not affect the r_{c} value much. Again, one can estimate β_{3}, thus r_{c} as well as r_{c}′ and (r_{c} + r_{c}′)/2, by multiple regression of DA, θ, S, and AOU data for deep waters from lower thermocline and below, using equation (4c).

Section | Latitude Interval | Station Number | θ, °C | r_{p} (−O_{2}/P) | r_{n} (−O_{2}/N) | r_{p}/r_{n} (N/P) | β_{3} | r_{c} (−O_{2}/C_{org}) | r_{p}/r_{c} (C_{org}/P) |
---|---|---|---|---|---|---|---|---|---|

- a
No bottom waters in the Atlantic Ocean were used in the model calculation, because the change of oxygen concentration in bottom waters can be explained solely by water mass mixing.
| |||||||||

Redfield | 138 | 8.6 | 16 | 0.83 | 1.30 | 106 | |||

a20 | 42°N–33.5°N | 12–33 | 3–18 | 142 ± 2 | 8.6 ± 0.1 | 16.5 ± 0.3 | 0.66 ± 0.02 | 1.67 ± 0.06 | 85 ± 2 |

33°N–13.5°N | 34–63 | 3–6 | 132 ± 2 | 8.5 ± 0.2 | 15.5 ± 0.4 | 0.58 ± 0.04 | 1.93 ± 0.15 | 68 ± 2 | |

6–12 | 145 ± 3 | 8.7 ± 0.2 | 16.7 ± 0.5 | 0.62 ± 0.02 | 1.79 ± 0.07 | 81 ± 3 | |||

13°N–7°N | 64–92 | 3–5.2 | 140 ± 3 | 9.6 ± 0.3 | 14.6 ± 0.6 | 0.52 ± 0.02 | 2.15 ± 0.10 | 65 ± 2 | |

a17 | 6°N–0° | 175–235 | 4.2–12 | 125 ± 3 | 7.9 ± 0.1 | 15.8 ± 0.4 | 0.64 ± 0.01 | 1.75 ± 0.04 | 72 ± 2 |

0°–25°S | 88–174 | 4–12 | 140 ± 2 | 8.5 ± 0.1 | 16.5 ± 0.3 | 0.67 ± 0.01 | 1.65 ± 0.03 | 85 ± 2 | |

25°S–40°S | 33–87 | 0.2–3 | 124 ± 3 | 8.0 ± 0.1 | 15.5 ± 0.4 | 0.74 ± 0.01 | 1.48 ± 0.03 | 84 ± 2 | |

3–12 | 127 ± 1 | 8.6 ± 0.1 | 14.8 ± 0.2 | 0.66 ± 0.01 | 1.67 ± 0.03 | 76 ± 1 | |||

40°S–50°S | 5–32 | 0.4–2.6 | 126 ± 4 | 8.5 ± 0.2 | 14.8 ± 0.6 | 0.74 ± 0.01 | 1.48 ± 0.03 | 85 ± 3 | |

2.6–12 | 126 ± 2 | 8.8 ± 0.2 | 14.3 ± 0.4 | 0.64 ± 0.01 | 1.73 ± 0.03 | 73 ± 2 | |||

average | 133 ± 8 | 8.6 ± 0.4 | 15.5 ± 0.8 | 0.65 ± 0.06 | 1.73 ± 0.19 | 77 ± 7 | |||

Geosecs | 42°N–12°N | 27–38 | 2–6 | 132 ± 5 | 7.9 ± 0.3 | 16.7 ± 0.9 | |||

6–18 | 127 ± 4 | 8.1 ± 0.2 | 15.7 ± 0.6 | ||||||

9°N–8°S | 39–49 | 1.8–3.8 | 116 ± 1 | 8.9 ± 0.1 | 13.0 ± 0.2 | 0.70 ± 0.06 | 1.56 ± 0.15 | 74 ± 1 | |

3.8–12 | 120 ± 3 | 8.8 ± 0.1 | 13.6 ± 0.4 | 0.71 ± 0.04 | 1.54 ± 0.10 | 78 ± 2 | |||

8.5°S–24°S | 50–57 | 1.8–4 | 121 ± 2 | 9.0 ± 0.1 | 13.4 ± 0.3 | ||||

3.2–12 | 133 ± 2 | 8.9 ± 0.1 | 14.9 ± 0.3 | 0.69 ± 0.02 | 1.59 ± 0.05 | 84 ± 2 | |||

27°S–49°S | 59–68 | 2.6–12 | 128 ± 1 | 9.0 ± 0.1 | 14.2 ± 0.2 | 0.71 ± 0.02 | 1.54 ± 0.05 | 83 ± 1 | |

average | 126 ± 6 | 8.8 ± 0.6 | 14.5 ± 1.2 | 0.70 ± 0.01 | 1.55 ± 0.02 | 82 ± 5 |

Section | Latitude Interval | Station Number | θ, °C | r_{p} (−O_{2}/P) | r_{n} (−O_{2}/N) | r_{p}/r_{n} (N/P) | β_{3} | r_{c} (−O_{2}/C_{org}) | r_{p}/r_{c} (C_{org/}P) |
---|---|---|---|---|---|---|---|---|---|

- a
Character “b” after a number in the θ column represents the potential temperature of bottom water. Without “b” means no bottom water samples were used in the model calculation. Symbols: *, θ ≥ (S − 34.74) × 3.8/0.24 + 1.8; **, θ ≥ (S − 34.72) × 4/0.12 + 2 in the θ-S diagram, i.e., samples above the Circumpolar Water.
| |||||||||

Redfield | 138 | 8.6 | 16 | 0.83 | 1.30 | 106 | |||

i8N | 6°N–0.5°S | 279–303 | 1b–9 | 120 ± 3 | 13.0 ± 0.3 | 9.2 ± 0.3 | 0.79 ± 0.04 | 1.34 ± 0.07 | 90 ± 3 |

0.5°S–18.5°S | 304–343 | 0.9b–8 | 129 ± 2 | 12.6 ± 0.4 | 10.2 ± 0.4 | 0.78 ± 0.04 | 1.36 ± 0.07 | 95 ± 3 | |

19°S–29.5°S | 344–393 | 2–8 | 129 ± 1 | 10.4 ± 0.1 | 12.4 ± 0.2 | 0.67 ± 0.03 | 1.62 ± 0.08 | 80 ± 1 | |

30°S–33°S | 394–442 | 2–8 | 127 ± 2 | 10.4 ± 0.2 | 12.2 ± 0.3 | 0.69 ± 0.07 | 1.57 ± 0.17 | 81 ± 2 | |

i8S | 30°S–51°S | 4–49 | 2–11 | 133 ± 3 | 9.5 ± 0.3 | 14.0 ± 0.5 | 0.66 ± 0.03 | 1.66 ± 0.08 | 80 ± 3 |

51°S–63.5°S | 50–84 | * | 130 ± 3 | 8.5 ± 0.2 | 15.3 ± 0.5 | 0.72 ± 0.03 | 1.52 ± 0.07 | 85 ± 3 | |

i9N | 20°N–2.5°N | 213–277 | 0.8b–11 | 131 ± 1 | 13.0 ± 0.2 | 10.1 ± 0.2 | 0.77 ± 0.02 | 1.37 ± 0.04 | 95 ± 2 |

2°N–18.5°S | 172–212 | 0.8b–8 | 142 ± 1 | 13.4 ± 0.2 | 10.6 ± 0.2 | 0.76 ± 0.02 | 1.39 ± 0.04 | 102 ± 2 | |

19°S–31°S | 148–171 | 2–10 | 134 ± 1 | 10.5 ± 0.2 | 12.8 ± 0.2 | 0.69 ± 0.01 | 1.57 ± 0.03 | 86 ± 1 | |

i9S | 34.5°S–50°S | 114–147 | 2–11 | 152 ± 2 | 9.8 ± 0.2 | 15.5 ± 0.4 | 0.60 ± 0.05 | 1.83 ± 0.17 | 83 ± 2 |

50°S–64.5°S | 85–113 | ** | 129 ± 2 | 8.5 ± 0.1 | 15.2 ± 0.3 | 0.70 ± 0.05 | 1.57 ± 0.12 | 82 ± 2 | |

Geosecs | 15°N–10°S | 440–450 | 0.7b–12 | 129 ± 2 | 11.5 ± 0.2 | 11.2 ± 0.3 | 0.73 ± 0.02 | 1.46 ± 0.04 | 88 ± 2 |

10°S–30°S | 436–439 | 0.4b–12 | 137 ± 1 | 10.2 ± 0.1 | 13.4 ± 0.2 | 0.71 ± 0.01 | 1.52 ± 0.02 | 90 ± 1 | |

39°S–63°S | 429–434 | 0–2.6 | 126 ± 2 | 9.0 ± 0.1 | 14.0 ± 0.3 |

Section | Latitude Interval | Station Number | θ, °C | r_{p} (−O_{2}/P) | r_{n} (−O_{2}/N) | r_{p}/r_{n} (N/P) | β_{3} | r_{c} (−O_{2}/C_{org}) | r_{p}/r_{c} (C_{org/}P) |
---|---|---|---|---|---|---|---|---|---|

- a
Character “b” in the θ column represents the potential temperature of bottom water. - b
Samples above the Circumpolar Water.
| |||||||||

Redfield | 138 | 8.6 | 16 | 0.83 | 1.30 | 106 | |||

p14N | 52°N–45°N | 20–34 | 1b–4.3 | 176 ± 4 | 13.1 ± 0.4 | 13.4 ± 0.5 | 0.76 ± 0.03 | 1.39 ± 0.06 | 127 ± 5 |

45°N–26.5°N | 35–72 | 1b–12 | 165 ± 3 | 12.0 ± 0.2 | 13.8 ± 0.3 | 0.70 ± 0.01 | 1.53 ± 0.03 | 108 ± 3 | |

26°N–11.5°N | 73–102 | 0.8b–6 | 162 ± 4 | 11.6 ± 0.2 | 14.0 ± 0.4 | 0.83 ± 0.02 | 1.28 ± 0.03 | 127 ± 4 | |

6–12 | 136 ± 1 | 12.0 ± 0.1 | 11.3 ± 0.1 | 0.82 ± 0.01 | 1.29 ± 0.02 | 105 ± 1 | |||

11°N–4°N | 103–126 | 0.8b–10.5 | 173 ± 6 | 12.7 ± 0.4 | 13.7 ± 0.6 | 0.85 ± 0.03 | 1.24 ± 0.05 | 140 ± 7 | |

p15N | 54°N–41°N | 2–31 | 1b–6 | 174 ± 2 | 12.8 ± 0.1 | 13.6 ± 0.2 | 0.71 ± 0.01 | 1.50 ± 0.02 | 116 ± 2 |

40°N–26.5°N | 32–59 | 1b–12 | 165 ± 4 | 12.5 ± 0.2 | 13.2 ± 0.4 | 0.71 ± 0.02 | 1.50 ± 0.05 | 110 ± 3 | |

26°N–12.5°N | 66–87 | 0.8b–6 | 152 ± 6 | 11.7 ± 0.4 | 13.0 ± 0.7 | 0.83 ± 0.03 | 1.28 ± 0.05 | 119 ± 6 | |

12°N–0° | 88–111 | 0.65b–12 | 170 ± 7 | 12.9 ± 0.4 | 13.2 ± 0.7 | 0.80 ± 0.03 | 1.32 ± 0.05 | 129 ± 7 | |

0°–12°S | 115–136 | 0.65b–6 | 165 ± 4 | 12.7 ± 0.1 | 13.0 ± 0.3 | 0.80 ± 0.02 | 1.32 ± 0.04 | 125 ± 5 | |

p15S | 9.5°S–15.5°S | 146–182 | 0.6b–2 | 173 ± 3 | 11.9 ± 0.4 | 14.5 ± 0.6 | 0.87 ± 0.05 | 1.21 ± 0.07 | 143 ± 5 |

2–12 | 128 ± 1 | 8.9 ± 0.1 | 14.4 ± 0.2 | 0.68 ± 0.01 | 1.61 ± 0.03 | 79 ± 1 | |||

16°S–42°S | 89–145 | 0.5b–2 | 158 ± 4 | 12.0 ± 0.3 | 13.2 ± 0.5 | 0.90 ± 0.08 | 1.17 ± 0.11 | 135 ± 5 | |

2–10 | 123 ± 3 | 9.2 ± 0.2 | 13.4 ± 0.4 | 0.62 ± 0.03 | 1.78 ± 0.10 | 69 ± 2 | |||

43°S–53°S | 61–87 | 1.8–11.3 | 148 ± 1 | 9.9 ± 0.1 | 14.9 ± 0.2 | 0.59 ± 0.01 | 1.87 ± 0.04 | 79 ± 1 | |

53°S–64°S | 36–60 | −1–8^{b} | 136 ± 1 | 9.2 ± 0.1 | 14.8 ± 0.2 | 0.62 ± 0.01 | 1.78 ± 0.03 | 76 ± 1 | |

p14S | 53°S–67°S | 4–32 | 1.7–7.6 | 136 ± 4 | 9.3 ± 0.2 | 14.6 ± 0.5 | 0.66 ± 0.02 | 1.66 ± 0.06 | 82 ± 3 |

P18 | 16°S–28.5°S | 82–107 | 1.5–12 | 123 ± 2 | 8.7 ± 0.2 | 14.1 ± 0.4 | 0.61 ± 0.02 | 1.82 ± 0.07 | 67 ± 2 |

29°S–40.5°S | 58–81 | 1.5–6.5 | 129 ± 2 | 10.1 ± 0.2 | 12.8 ± 0.3 | 0.63 ± 0.01 | 1.73 ± 0.03 | 74 ± 2 | |

54.5°S–67°S | 10–33 | 2–7 | 131 ± 2 | 8.8 ± 0.3 | 14.9 ± 0.6 | 0.66 ± 0.03 | 1.67 ± 0.09 | 79 ± 3 | |

Geosecs | 50°N–40°N | 1b–8 | 176 ± 1 | 11.6 ± 0.1 | 15.2 ± 0.2 | 0.74 ± 0.02 | 1.44 ± 0.04 | 122 ± 1 | |

35°N–25.5°N | 0.9b–12 | 158 ± 2 | 12.1 ± 0.1 | 13.1 ± 0.2 | 0.80 ± 0.03 | 1.32 ± 0.05 | 119 ± 2 | ||

25°N–10°N | 0.8b–6 | 152 ± 4 | 11.6 ± 0.3 | 13.1 ± 0.5 | |||||

9°N–13.5°S | 0.6b–12 | 159 ± 5 | 11.8 ± 0.2 | 13.5 ± 0.5 | 0.83 ± 0.05 | 1.27 ± 0.08 | 125 ± 4 | ||

15°S–42°S | 2–12 | 152 ± 5 | 10.0 ± 0.2 | 15.2 ± 0.6 | |||||

46.5°S–53°S | 2–12 | 137 ± 3 | 9.9 ± 0.3 | 13.8 ± 0.5 | |||||

54°S–69°S | 2–6 | 148 ± 3 | 8.8 ± 0.3 | 16.8 ± 0.7 | 0.70 ± 0.07 | 1.56 ± 0.17 | 95 ± 4 |

[15] One needs not assume O_{2}^{0} to be equal to the saturated oxygen in equilibrium with air, and rewrite equation (4c) to

Substituting O_{2}^{0} = f_{1} · (O_{2}^{0})_{1} + f_{2} · (O_{2}^{0})_{2} + f_{3} · (O_{2}^{0})_{3} into equation (4d), one obtains

where constants B_{0}, B_{1}, and B_{2} are bulky functions of θ_{i}^{0}, S_{i}^{0}, DA_{i}^{0}, O_{2}^{0}_{i}, for three end-members and β_{3}.

[16] By multiple regression of DA, θ, S, and O_{2} data using equation (4e), one can estimate β_{3}, thus r_{c}, as well as r_{c}′ and (r_{c} + r_{c}′)/2. In practice, both equations (4c) and (4e) give the same β_{3} values within the estimated uncertainty. Therefore, the assumption that O_{2}^{0} represents the saturated oxygen in equilibrium with air is a good approximation.

[17] In short, the advantage of the present three-end-member model is that the data points selected for the model need not be confined in an isopycnal horizon [*Takahashi et al.*, 1985; *Broecker et al.*, 1985; *Minster and Boulahdid*, 1987] or neutral surface [*Anderson and Sarmiento*, 1994]. The data points can be expanded greatly (easily to more than one hundred to several hundreds data points for each run) to include those in a vertical cross section or in a large volume of a given regional basin without having to worry about distinguishing horizontal/vertical or isopycnal/diapycnal mixing.