Reconciliation of pH25 and pHinsitu acidification rates of the surface oceans: A simple conversion using only in situ temperature

Seawater pH is frequently measured at 25°C (pH25), and can be converted thermodynamically to pH at the in situ temperature (T), (pHinsitu) using an additional carbonate chemistry parameter, which is the total alkalinity (TA), dissolved inorganic carbon (DIC), or the partial pressure of CO2 (pCO2) of seawater. Although rates of temporal change of pHinsitu ( βpHinsitu ) and pH25 ( βpH25 ) are both extensively used in studies of ocean acidification, the difference between βpHinsitu and βpH25 has not yet been quantified. This study deducts from 816 sets of data of the surface oceans over wide ranges of T (1–31°C) from six time series to reveal that the difference between calculated pHinsitu and pH25 is a1 (T − 25°C), where a1 is a nearly constant of −0.0151 pH unit °C−1. We illustrate that βpHinsitu equals ( βpH25  + a1 βT ), where βT is the rate of temporal change of T. We further show that uneven distributions of sampling points significantly widen the difference between βpHinsitu and βpH25 , making the degree of ocean acidification unclear. Distributions of a1 values are modeled for the surfaces of the global oceans at various pCO2 levels, and they closely match the observations from the studied time series. Without the use of an additional carbonate chemistry parameter, the pHinsitu and pH25, as well as βpHinsitu and βpH25 can now be converted into each other using only T, facilitating the study of the changing carbonate chemistry of seawater under the influences of increasing atmospheric CO2 concentration.

The pH of seawater reflects directly the state of the acidbase systems of the oceans (Marion et al. 2011). It has attracted much attention recently, as it reflects the seawater acidification of the oceans under the influence of the increasing atmospheric CO 2 concentration (Dore et al. 2009;Olafsson et al. 2009;Byrne et al. 2010;Gonzalez-Davila et al. 2010;Ishii et al. 2011;Bates et al. 2014;, which has in turn been caused by the fact that since the industrial revolution, humans have released a massive amount of CO 2 , so-called anthropogenic CO 2 , to the atmosphere. The global oceans absorb around one third of the anthropogenic CO 2 , increasing their CO 2 concentration but reducing their pH and the saturation state of calcium carbonate through the air-sea CO 2 exchange, adversely affecting marine ecosystems (Sabine et al. 2004;Dore et al. 2009;Feely et al. 2009;Olafsson et al. 2009;Bates et al. 2014).
The dissolution of CO 2 and the dissociation constants, K 0 , K 1 , and K 2 , can be expressed as follows.  (Ben-Yaakov 1970;Hunter 1998;Orr et al. 2015). Since the dissociation constants are functions of seawater temperature (T) and salinity (S), a change in T or S alters the speciation of the carbonate system and therefore the pH value (Ben-Yaakov 1970;Mehrbach et al. 1973;Dickson and Millero 1987;Millero 1995). pH is commonly measured at a constant temperature, such as at the most favored 258C (pH 25 ). The pH 25 value can then be converted thermodynamically to pH at the in situ T (pH insitu ) when the DIC, TA, or pCO 2 of seawater has been measured.
The pH 25 and pH insitu time series have been used to determine how the oceans respond to the increase in anthropogenic CO 2 concentration. Traditionally, the simple-linearregression (SLR) method has been applied to the pH insitu or pH 25 time series. The slopes of the regression lines of the pH insitu (b pHinsitu ) and pH 25 (b pH25 ) time series reflect the rate of temporal change of pH, which is the so-called acidification rate (Dore et al. 2009;Gonzalez-Davila et al. 2010;Midorikawa et al. 2010;Ishii et al. 2011;Bates et al. 2014;. However, as shall been shown later, the reported b pHinsitu and b pH25 may differ significantly. For example, at the Carbon Retention in a Colored Ocean Project (CARIACO) site, reported b pHinsitu values are between 20.00214 and 20.0025 pH unit yr 21 (Astor et al. 2013;Bates et al. 2014), which are about 56% higher than the expected rate of 20.0017 pH unit yr 21 , assuming air-sea CO 2 equilibrium. In contrast, the b pH 25 is just 20.0004 pH unit yr 21 (Astor et al. 2013), which is 76% lower than obtained by assuming air-sea CO 2 equilibrium. The difference between b pHinsitu and b pH25 , however, has not yet been quantified so reported b pHinsitu and b pH25 values are incomparable, making the degree of ocean acidification unclear.
Thermodynamically, the conversion between pH 25 and pH insitu is a non-linear function of T, S, and an additional value of TA, DIC, or pCO 2 . This study uses 816 time series measurements of the surface seawaters from six stations in the global oceans to show that the difference between calculated pH insitu and pH 25 is indeed basically a linear function of T. Furthermore, the difference between b pHinsitu and b pH 25 is shown to be a linear function of the rate of temporal change of T (b T ). The difference between b pHinsitu and b pH25 is shown potentially to be increased by the uneven distributions of the sampling points of the time series. The implications of our findings for studies of changing seawater carbonate systems are discussed.

Methods and materials
In this study, time series data from six stations in the global surface oceans are analyzed. The stations include the Bermuda Atlantic Time Series Study (BATS, taken from Bates (2007) and http://bats.bios.edu), CARIACO (taken from Astor et al. (2013) (Fig. 1). The up-to-date CO 2 System Calculations Program version 2.1, developed by Pierrot et al. (2006), is used to calculate the measureable carbonate chemistry parameters, using recommended dissociation constants for carbonate chemistry that are taken from Lueker et al. (2000) (Dickson et al. 2007;Orr et al. 2015). The pH insitu at the CARIACO are calculated using the measured pH 25 and TA data, and the pH insitu and pH 25 at the other stations are calculated using the measured TA and DIC data. The b pHinsitu , b pH 25 ; and b T values are shown in Table 1.
To study the distributions of the changes in pH insitu with T (a 1 ) at the surfaces of the global oceans, the climatological monthly T, S from the World Ocean Atlas 2001, and the TA data (31,607 data points) estimated using the monthly T and S of the surface oceans, presented in Lee et al. (2006), are used to calculate the pH insitu and pH 25 at various pCO 2 values using the MATLAB Program developed for CO 2 system calculations (version 1.1) (van Heuven et al. 2011). The phosphate and silicate concentrations are assumed to be zero. At each station, the climatological monthly differences between pH insitu and pH 25 (DpH insitu-25 ) values are plotted vs. the monthly T, and a 1 is then obtained using the SLR method by forcing the regression line to pass through the reference point (T 5 258C, DpH insitu-25 5 0). The pH insitu and pH 25 are reported in the total scale. The values in this study are expressed as value 6 one standard error.

Results and discussion
Conversions between pH insitu and pH 25 and between b pH insitu and b pH 25 using T Figure 2 plots pH insitu minus pH 25 (DpH insitu-25 ) vs. T at six time series stations. Interestingly, although the seawaters at the studied time series stations have wide ranges of T (1-318C), pH insitu (8.002-8.189) and pH 25 (7.656-8.130), they exhibit almost identical DpH insitu-25 vs. T linearity. pH 25 equals pH insitu when T is at 258C. The regression reveals that the SLR lines have almost the same slopes, between 20.0150 and 20.0152 pH unit 8C 21 , when they are forced to pass through the reference point at DpH insitu-25 5 0 and T 5 258C. The average standard error is just 6 0.00036 pH unit (or just 6 0.0001 pH unit when the KNOT data is excluded) ( Table 2). The slopes are consistent with DeGrandpre et al.
(2014)'s calculations of 20.015 pH unit 8C 21 at the HOT and BATS stations and 20.016 pH unit 8C 21 for the coastal water in the northeast Pacific Ocean, suggesting the ranges of the slopes (the a 1 values) of the surface world oceans are fairly narrow as will be shown and discussed later.
As mentioned, the conversion between pH 25 and pH insitu is a non-linear function of T, S, and an additional value of TA, DIC, or pCO 2 . Empirically, pH insitu can be expressed as a nonlinear function of pH 25 , T, S, and the TA/DIC ratio (Millero 1995). In past decades, the TA/DIC ratio has declined insignificantly in response to the increase in atmospheric CO 2 concentration. For example, at the HOT site, the TA/DIC ratio has declined by just about 1% during the studied period (Dore et al. 2009). Additionally, the effect of S on the conversion is also insignificant when S changes only a little (by less than 5, for example) (Millero 1995). Since S values in most parts of the oceans around the world, especially in the case of time series data, vary in a narrow range, the conversion depends on the use of only T. Based on the result above, the pH insitu and pH 25 among the studied time series can be written as follows.
where a 1 is the slope of the plot of DpH insitu-25 vs. T, referring to the amount of pH insitu change as T increases. Based on Eq. 1, the conversion between the long-term trends of pH insitu and pH 25 can be simplified using a factor a 1 b T , and is discussed as follows. To determine the rate of acidification of the oceans, the SLR method has been used to model the long-term temporal changes of pH insitu and pH 25 . For any pH insitu time series, b pHinsitu is defined as follows (Montgomery et al. 2006).
where pH insituðiÞ and t i are pH insitu and time (t), respectively at t5t i , and t is the average t. Substituting Eq. 1 into Eq. 2 yields, b T (see the definition shown in Eq. 2) and X n i51 t i 2 t ð Þ50, the above equation can be simplified as follows.
Equation 3 reveals that, mathematically, when a 1 is a constant, the difference between b pHinsitu and b pH 25 is a 1 b T . Generally, Eq. 1 approximates the thermodynamic effect of T on pH insitu , and is not affected by biological activities. Therefore, the transformation between b pHinsitu and b pH 25 , the Eq. 3, is also a thermodynamic result, albeit complicated by biological activities. To confirm Eq. 3, observations from six time series with various numbers of observation years are examined with the averages of 31 time series stations along the 137 o E hydrographic line (137 o E) in the winter and summer (Table 1).  noted that the actual longterm temporal changes in time series of pH insitu and T data can be easily masked by strong seasonal variations, especially when the sampling distribution is uneven. Using the same dataset as , this study determined that the b pH 25 at the ESTOC between 1995 and 2009 is 20.0015 6 0.0002 pH unit yr 21 . The b pH 25 would have been 20.0025 6 0.0004 pH unit yr 21 (Table 1), with a selected ESTOC dataset which has a sampling time that gradually shifts from summer to winter (for detail see . Same for b pHinsitu and b T , the unevenly distributed ESTOC dataset differ significantly from those from the complete dataset (Table 1). Figure 3 plots b pHinsitu vs. b pH 25 1 a 1 b T . The result shows that all data, except for those at the CARIACO (deseasoned) and the ESTOC sites, fall on the 1 : 1 line, validating our proposed Eq. 3. The deviations at the CARIACO and ESTOC sites arise from statistical errors since their T time series do not have the same sampling distributions as pH insitu and pH 25 . Based on Eq. 3, a direct comparison between b pHinsitu and b pH 25 yields a difference of a 1 b T . Worth mentioning, the datasets used in this study cover wide ranges of T (1-318C), pH insitu (8.002-8.189), pH 25 (7.656-8.130), TA (2167-2673 lmol kg 21 ), DIC (1859-2389 lmol kg 21 ), as well as normalized TA (NTA 5 TA/S 3 35, 2252-2534 lmol kg 21 ) and DIC (NDIC 5 DIC/S 3 35, 1928-2251 lmol kg 21 ) values. Although five of the time series stations have stable NTA concentrations, those at CARIACO show large temporal variations (2252-2534 lmol kg 21 ). That is, our proposed Eqs. 1 and 3 are applicable over a wide T, pH insitu , pH 25 , NTA (or TA), and NDIC (or DIC) ranges. The distributions of a 1 values of the surface oceans will be shown and discussed later.
Uneven distributions of sampling points increase differences between b pH insitu and b pH 25 From Eq. 3, the difference between b pHinsitu and b pH 25 is a 1 b T , where a 1 is almost a constant and is between 20.0150 and 20.0152 pH unit 8C 21 in the six time series (Fig. 2). That is, the difference between b pHinsitu and b pH 25 increases with increasing b T . Using ESTOC as an example,  showed an extreme example of uneven sampling distributions that the sampling time gradually shifts from summer to winter. The result was that the b T changed from 0.023 6 0.0398C yr 21 (complete dataset) to 20.273 6 0.0368C yr 21 (Table 1). Figure 4 shows that although the observed b pHinsitu and b pH 25 at the six stations and 137 o E line varied greatly, the differences between b pHinsitu and b pH 25 are negatively correlated with b T . All data, except for those at the CARIACO (deseasoned) and the ESTOC sites, fall on the 20.0151b T line, and the 20.0151 pH 8C 21 is the average a 1 value among the six time series (Table 2). As mentioned, the slight deviations at the CARIACO and ESTOC sites arise from statistical   (Table 2). errors since their T time series do not have the same sampling distributions as pH insitu and pH 25 . Generally speaking, the b T that is caused by global warming is only about 20.018C yr 21 (Karl et al. 2015), so the large range of the observed b T values is caused largely by the uneven sampling distribution. The above illustrates that uneven distributions of sampling points increase the differences between b pHinsitu and b pH 25 . In that case, either b pHinsitu and b pH 25 contain the deviations due to the uneven sampling distributions. Using our proposed method, pH insitu and pH 25 data, as well as their rates of temporal changes now can transform to each other using only T. This helps avoiding the confusion in evaluating the acidification rate using pH insitu or pH 25 data. We suggest that long-term monthly or seasonal sampling strategy efficiently helps reducing the deviations of b pHinsitu and b pH25 due to the uneven sampling distributions. In the case of short-term and unevenly distributed time-series data, the use of an appropriate regression model may help reducing the deviations. In the cases that pH insitu and pH 25 time series can be expressed empirically as functions of t and T, the deviations of b pHinsitu and b pH 25 due to the uneven sampling distributions can be largely removed using the multiple linear regression method with t and T as variables .
Global distributions and estimations of a 1 Figure 5 plots the distributions of the modeled a 1 values using the climatological monthly T (22 to 328C), S (31-38) and estimated TA data (2053-2494 lmol kg 21 ), taken from Lee et al. (2006), for pCO 2 values of 280 (pH insitu : 8.139-8.188, pH 25 : 7.739-8.260), 400 (pH insitu : 8.004-8.063, pH 25 : 7.610-8.140) and 800 (pH insitu : 7.720-7.813, pH 25 : 7.362-7.891) latm. The use of different color scales is to show the small regional differences of the a 1 values at various pCO 2 levels. The results reveal that although the surface ocean has various physical and chemical properties, it has very similar a 1 values, ranging from 20.01522 to 20.01537 (average 20.01529 6 0.00003) pH unit 8C 21 , when pCO 2 5 280 latm. The a 1 values decrease and the regional differences of a 1 values increase as pCO 2 increases. When pCO 2 is 400 latm, the a 1 values are slightly lower, at between 20.01510 and 20.01482 (average: 20.01502 6 0.00005) pH unit 8C 21 . At a pCO 2 of 800 latm, the a 1 values are between 20.01456 and 20.01357 (average 20.0141 6 0.000285) pH unit 8C 21 . The observed a 1 values of 20.01504 to 20.01519 pH unit 8C 21 , shown in Fig. 2, match closely the modeled a 1 values when pCO 2 is between 280 latm and 400 latm, validating the modeled results in Fig. 5. Generally speaking, the a 1 value is a function of T, S and a pair of TA, DIC, pCO 2 , and pH insitu or pH 25 . Therefore, to state the applicable ranges of each carbonate parameter for our proposed method is complicated. The above information provides a reference of their applicable ranges.
Although a 1 declines as pCO 2 increases, the value changes little as pCO 2 varies between pre-industrial and present levels. As shown in Fig. 5 and Table 2, a 1 is approximately 20.0151 between 50 o N and 50 o S, and it is about 20.0150 pH unit 8C 21 at higher latitudes. Notably, based on the assumption that the uncertainty in a 1 is 6 0.00036 pH unit 8C 21 (or just 6 0.0001 when KNOT is excluded) pH unit 8C 21 , a transformation of a pH 25 value to pH insitu at T as low as 08C yields an uncertainty of only (6 0.00036 3 25) 5 6 0.009 pH unit. The uncertainty is less when T is closer to 258C. Therefore, even lacking an additional carbonate parameter, historical pH insitu or pH 25 data for the surface ocean can still be converted to each other using only T and a 1 , which has been shown to be close to 20.0151 pH unit TA data (total of 31,607 points, taken from Lee et al. (2006)), and modeled a 1 values when pCO 2 is (b) 280, (c) 400, and (d) 800 latm. The use of different color scales is to show the small regional differences of a 1 values at various pCO 2 levels.
8C 21 . When the K 1 and K 2 values that are taken from Lueker et al. (2000) are used, the average a 1 of the studied time series is 20.01511 6 0.00006 pH unit 8C 21 . When the corresponding values from Mehrbach et al. (1973), refitted by Dickson and Millero (1987), and from Millero (2010), are used, a 1 becomes 20.01492 6 0.0001 and 20.01492 6 0.00007 pH unit 8C 21 , respectively. The differences between such values are approximately 0.0002 pH unit 8C 21 . Therefore, equilibrium constants should be used consistently in all calculations to prevent an additional, albeit small, systematic error.

Conclusions
The conversion between pH 25 and pH insitu traditionally, requires an additional carbonate parameter. This study reveals that only T and a coefficient, a 1 which is about 20.0151 pH unit 8C 21 , are required to convert linearly pH 25 and b pH 25 to pH insitu and b pH insitu , respectively, and vice versa. This study demonstrates that the difference between b pH insitu and b pH 25 can be significantly enlarged owing to uneven distributions of sampling points. Our method is applicable over wide T, pH insitu , pH 25 and NTA ranges, facilitating the study of the changing carbonate chemistry of seawater, such as to avoid the confusion in evaluating the acidification rate using pH insitu or pH 25 data.