A Mathematical Explanation for Why Ratio-Based Isotopic Analyses are Commonly Misleading: Dealing with Confounded Isotopic Ratios

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Nguyen, Catherine Nguyen, Frank Camacho, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4086527/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Dozens of preliminary data reevaluations were conducted to verify the ratio-related mathematical theory. Differences in total elements among treatments, times and/or conditions frequently confound interpretation because total element values affect isotopic ratios. Eventually, twelve (six 87 Sr: 86 Sr, three 15 N: 14 N, two 13 C: 12 C and one 34 S: 32 S) well-performed studies were selected as examples. Sr studies: Source evaluations better describe migration patterns for ancient humans and animals, better align speleothem isotopic data with known climate changes, better define the dynamics of isotopic data within a watershed, and better describe sources of soil Sr. N studies: Source evaluations change interpretations for isotopic fractionation in sediments; N tracer treatments on potted plants; and trophic level assignments for different species in a marsh. C studies: Total C confounds 13 C: 12 C data for isotopic fractionation experiments in forest soils and complicates an evaluation of whether past life existed in Martian sediments collected by the Curiosity rover. S studies : Total S also confounds 34 S: 32 S evaluations of the same Martian sediments. We intend to emphasize that source analyses provide better isotopic interpretations than observed ratios in agricultural, biological and environmental studies. Observed isotopic ratio changes do not necessarily reflect source changes. Source analyses improved the Sr, N, C and S isotope evaluations. Biological sciences/Biological techniques Earth and environmental sciences/Biogeochemistry Earth and environmental sciences/Climate sciences Earth and environmental sciences/Ecology Earth and environmental sciences/Environmental sciences Physical sciences/Chemistry Figures Figure 1 Figure 2 1. Introduction Isotopic researchers often assume that observed isotopic ratios reflect the source (isotopic ratio of material gained or lost) of a system. However, mathematical principles suggest that observed ratios in isotopic samples almost always differ from those of sources 1 . Observed ratios when analyzed without evaluating sources can be misleading 1 . Issues related to ratio use in scientific research 2–7 have been ignored for many decades 4,8 . We advocate using isotopic denominator vs. numerator plots to define isotopic sources. Keeling plots 9 and mixing diagrams 10–13 have been accepted approaches for identifying isotopic sources for more than 60 years. However, purely mathematical concepts reveal that Keeling plots and isotopic mixing diagrams are indirect ways to calculate the slope and relative y -intercepts of linear denominators vs. numerator isotope plots 1 . The only way in which a zero y -intercept for a linear denominator vs. numerator isotope function can mathematically occur is when sources and backgrounds are similar, which produces flat mixing diagrams and constant isotopic ratios 1 . Therefore, nonzero y -intercepts for linear denominator vs. numerator concentration plots are common, mathematically dictating that isotopic ratios are often related to their denominator or total elemental concentration 1 . Nonlinear relationships between denominator vs. numerator isotope concentrations affect how isotopic ratios change with increasing total element concentration in ways that are intuitively difficult to predict 1 . Total element concentrations, necessary for source analyses, are present in less than half of the isotopic publications 1 . Without evaluating the total elemental content, the relative background and source ratios cannot be determined. Even when total element data are available, researchers rarely conduct source analyses 1 . This approach is unfortunate because determining sources solves most interpretive issues. Derivatives ( d numerator/ d denominator) define changing sources. Nonzero y -intercepts quantify backgrounds for linear and curvilinear polynomial functions 1 . The portion of an element derived from an exogenous source is traditionally calculated from isotopic ratios 1 . Isotopic preferences (fractionation or discrimination factors) are also derived from isotopic ratios. Therefore, both of these isotopic estimates are subject to the same confounding scaling effects as the original ratios. A paradigm shift from the use of simple observed isotopic ratios to procedures that directly identify source, background and total isotope effects on an isotopic ratio is necessary. 2. Results: Dealing with Confounded Isotopic Ratios 2.1. Untangling Confounded Data: A Sample Analysis of a Constant Source Artificial Example Soil nutrients are generally thought to be derived primarily from rock weathering. This first example was calculated by modifying data from a study 14 in which Sr isotope data were collected at several depths on 30,000-year-old Pahala Ash deposits 50 m from the coast at South Point, Hawaii. The numbers adjacent to the hypothetical 87 Sr: 86 Sr ratio points in Fig. 2 a for different soil depths are calculated as percentages (using standard formulas) of the Sr at a given depth that originated from seawater spray added to the soil surface. Similar portions can be estimated by the relative position of a point between two possible sources (basalt and sea spray). A typical interpretation based on 87 Sr: 86 Sr ratios alone would suggest that seawater is an important source of Sr and that this influence decreases with depth. However, standard formulas used to calculate the portion of total elements derived from a suspected source do not account for the fact that the isotopic ratios used in these formulas are often confounded by nonzero y -intercepts, which make isotopic ratios dependent on total element concentrations. It is possible that the Sr additions to the soil in this artificial example could be due to a blending of both seawater (0.7091) and basalt (0.7035) 87 Sr: 86 Sr sources, with the relative amounts of input from the two sources varying with depth. The value for the basalt source stated above and shown in Fig. 1 a is for the basalt soil parent material suggested by the original authors. The original data can be interpreted differently depending on whether constant source or changing source analyses are conducted. To simplify this first data evaluation, the changing source explanation is much less likely for the modified data because a constant source (linear 86 Sr vs. 87 Sr relationship) adequately explains the results. In addition to demonstrating an alternative to ratio-based interpretations by providing examples of the five proposed ways of dealing with confounded ratios (Section 4.1 ), this modified dataset also reveals how mathematical principles can produce surprising results. Even if the observed isotopic ratios fall within the range defined by suspected sources (Fig. 1 a), the true source can be very different than either logically suspected source. A full analysis of the original unmodified data is included in the Supplementary Material, Section 2 . Surprisingly, although total Sr and total Ca were measured, these data are not presented in Whipkey et al . 14 Total Sr values were indirectly estimated before being modified for this example. The data were derived by first calculating the total Ca concentrations from the Na concentrations and molar Ca:Na ratios provided in the authors’ original publication. Next, total Sr was calculated from the derived Ca concentrations and molar Sr:Ca ratios. Once the total Sr concentration was known, the umole/g values for 86 Sr and 87 Sr could be calculated from the given 87 Sr: 86 Sr molar ratios using the assumption that the combined total portion of 86 Sr and 87 Sr was 9.8566% + 7.0015. These two values are the respective portions of the total values for 86 Sr and 87 Sr for the commonly used NBS SRM 987 Strontium Carbonate reference standard (used in all six Sr studies that were re-evaluated). Although, strictly speaking, concentration units are reserved for variables with units of volume, we use the broader definition commonly used in soils, sediments, and plant tissues (for example, µg/g or µmol/g). This modified dataset demonstrates how less than perfect linear relationships with levels of variability similar to what is encountered in real data can produce 87 Sr: 86 Sr ratios that are driven by changes in total Sr (or the highly correlated 86 Sr). This dataset also serves as an introduction to the fact that the denominator vs. numerator r 2 values presented in isotopic studies are spectacularly high. The data were altered to create a subtle concave curvature (polynomial x 2 term p = 0.14), which was not statistically significant. The values for total Sr and 86 Sr have also been modified to always increase with depth (unlike the original data). Furthermore, the original range of observed 87 Sr: 86 Sr ratios has been expanded to create a larger range between two logical but not necessarially accurate endmember sources (seawater and basalt). The relationship between 86 Sr and 87 Sr for the modified data presented in Fig. 1 a is presented in Fig. 1 b. The function is linear, and the y -intercept, although very small, is statistically significant ( p = 0.00012). It is difficult to determine how a small y -intercept (approximately 0.485% of the y value range) for a linear function could have interpretive consequences. However, mathematically, the ratio associated with a linear 86 Sr vs. 87 Sr function with a nonzero y -intercept must change with increasing denominator size ( y = m x + b dictates y / x = (1/ x ) b + m). This is apparent in Fig. 1 c, where the 87 Sr: 86 Sr ratio decreases with increasing 86 Sr, as predicted from the original 86 Sr vs. 87 Sr linear function with a positive y -intercept shown in Fig. 1 b. The r 2 values for both linear and polynomial ( x 2 term not significant) best-fit equations are very high (Fig. 1 a) but in the range of what one often encounters in real data 1 (see Supplementary Materials 1–3 in reference 1). Differences in total Sr or strongly correlated 86 Sr, both of which change with depth, drove the decrease in 87 Sr: 86 Sr ratios rather than changes in exogenous sources (Fig. 1 c). The 86 Sr vs. 87 Sr linear slope of 0.698532 (an indicator of an exogenous source) is significantly lower (99% confidence intervals of 0.69799–0.699066) than that of seawater (~ 0.7091) or basalt (0.7035), suggesting that neither sea spray nor basalt is an important exogenous source. Source inference based on 87 Sr: 86 Sr ratios is likely incorrect. The 1/denominator form of a mixing diagram where 1/ 86 Sr is plotted against the observed and y -intercept adjusted ratios is shown in Fig. 1 d. Figure 1 d also shows a slightly changing spline smoothed cubic derivative (small gray points) and the specific 87 Sr: 86 Sr derivative values (large gray points) associated with known 86 Sr values at the five depths. This change in derivative (increasing with increasing 1Sr/ 86 Sr or decreasing with increasing 86 Sr) is what one would expect from the subtle but not statistically significant difference in the 86 Sr vs. 87 Sr plots in Fig. 1 b. However, for an evaluation of the source, the relationship between 86 Sr and 87 Sr is essentially linear. The slope of the mixing line (0.00000007105) in Fig. 1 d approximates the y -intercept (0.000000071786) of the original 86 Sr vs. 87 Sr plot shown in Fig. 1 b. The y -intercept of this mixing line (0.69859) closely matches the slope (0.69853) shown in Fig. 1 b. Therefore, either a mixing diagram or an 86 Sr vs. 87 Sr plot can be used to identify exogenous sources and relative differences in backgrounds. A mixing diagram constructed from y -intercept adjusted points is also shown in Fig. 1 d. The y -intercept adjustments were made as described above (Section 3.1 and Supplementary Material 1 in ref 1). This process eliminates y -intercept effects and produces a flat mixing line if the original 86 Sr vs. 87 Sr plot is indeed linear. In this case, the flattened mixing line produces an almost identical source ( y -intercept) as the original mixing line. Individual points on this flattened mixing line represent the source estimates for individual depths. Additionally, Fig. 1 d shows the spline smoothed cubic derivative, which includes the d 87 Sr/ d 86 Sr derivative values associated with known 86 Sr values at five depths. This change in derivative (increasing with increasing 1/ 86 Sr or decreasing with increasing 86 Sr) is what one would expect from the subtle but not statistically significant concave curvature in Fig. 1 b. All three source estimates in Fig. 1 d (slope, modified mixing line with 1/ 86 Sr on the y -axis, and a similar y -intercept modified ratio mixing line) are in agreement. A traditional 1/total Sr mixing line also produces a similar source estimate (0.698594 data not shown). All five alternate source assessments shown in Fig. 1 e (traditional 1/total Sr mixing line ( x -axis; µg/g units); 1/ 86 Sr form of a mixing line ( x -axis; µgmole/g units); y -intercept corrected mixing line; 86 Sr vs. 87 Sr slope; and spline smoothed cubic derivative) plot against soil depth. Seawater is an unlikely source. The portion of Sr derived from the seawater calculations presented in Fig. 1 a is based on 87 Sr: 86 Sr ratios confounded by nonzero y -intercepts. One might suspect that the derivative analysis in Fig. 1 e, which detected the subtle but not statistically significant concave 86 Sr vs. 87 Sr relationship curvature, was the most accurate. The 86 Sr vs. 87 Sr relationship is not perfectly described by a linear function. The small x -axis scale reveals apparent source pattern differences with soil depth that are likely unimportant. In Fig. 1 f, the five different source estimates (all shown in Fig. 1 e) are plotted vs. depth along with the original 87 Sr: 86 Sr observed ratios, which require an expanded x -axis scale. On the expanded scale, differences among the five source estimates are indeed unimportant. Seawater spray was clearly not the source of Sr in this example. An unknown exogenous material with a smaller 87 Sr: 86 Sr ratio than the basalt parent material must be the source of these artificial data. Although not crucial in interpreting this artificial example, it is prudent to investigate whether apparently linear 86 Sr vs. 87 Sr relationships might have subtle curvatures. In the original unmodified dataset (Supplementary Material 1), possible curvatures in the 86 Sr vs. 87 Sr relationship are likely interpretively important. In some studies, changes in an observed isotopic ratio are likely due to changes in the isotopic ratio of material being added or removed from a system with different initial isotopic compositions. However, whether this is true cannot be determined in most isotopic research reports because they do not include total element data. 2.2. Untangling Confounded Data: A Sample Analysis of a More Complex Changing Source Real Data Example The main objective of our first changing source example 15 was to investigate whether and how a speleothem-derived 87 Sr: 86 Sr ratio and high-resolution total Sr record are correlated with the past climate and environment in the East Asian summer monsoon (EASM) region during the great climate shift associated with the last deglaciation. Samples were collected from a speleothem (stalagmite) in Songjiang Cave, Northeast Sichuan Province, Central China. Zhou et al . 15 present cave data from central China for total Sr at various stalagmite depths that represent deposits between 20,000 and 10,000 years before present. We separated the data into three different time periods corresponding to three different depths (10–12.25 shallow; 12.5–14.25 deep; and 14.25–20 thousand years deep before present), as shown in Fig. 2 a. Changes in total Sr could drive differences in 87 Sr: 86 Sr ratios. If 86 Sr vs. 87 Sr relationships have significant curvature and/or significant nonzero y -intercepts, 87 Sr: 86 Sr ratios can vary with total Sr in ways that are counterintuitive and difficult to predict 1 . The 87 Sr: 86 Sr ratios for the same depths are presented in Fig. 2 b. These time divisions are based on stable 18 O: 16 O ratios that correspond to the ice core layer dates described below. The original authors include examples of 18 O data from several speleothems 16–18 and compare these patterns to samples from Greenland ice cores. Temporal patterns differ. One would predict that different speleothem samples would differ in total O content since CaCO 3 concentrations and other sources of O differ for different speleothem layers at a single location and for similar times at different locations. Thus, speleothem 18 O: 16 O ratios that are confounded by total O differences will produce temporal patterns that differ. However, ice core samples will have constant O concentrations for different layers or sites because the O content of water is always the same. There is no scaling of isotopic ratios if the denominators for different samples have constant total elemental concentrations. The ice core data copied from the Zhou et al . 15 publication (Fig. 2 c) reveal a decrease followed by an increase followed by stable 18 O: 16 O ratios that correspond to generally accepted changes in climate that are associated with the ice core layer dates. The original authors suggest simple binary mixing of two endmembers, the host rock of late Permian limestone, with a relatively lower 87 Sr: 86 Sr ratio (∼0.7071) and an exogenous Sr source with a relatively high 87 Sr: 86 Sr ratio (∼0.7109 deduced from 1/total Sr ( x -axis) vs. 87 Sr: 86 Sr ratio ( y -axis) plots). We hypothesize that although mixing diagram equations similar to previously published results can be derived from a 1/ 86 Sr vs. 87 Sr: 86 Sr ratio plot, as shown in Fig. 2 d (upper left), there are likely three different sources and backgrounds (also shown in the upper right in Fig. 2 d) for the different depths that correspond to the time periods and climate differences revealed by changes in ice core oxygen isotopes (also shown in Fig. 2 d). The nonlinear mixing lines (Fig. 2 d) for the shallow and transition depths suggest that the sources for these two depths are not constant. The shallow pattern is as expected for concave 86 Sr vs. 87 Sr relationships with negative y -intercepts 1 . The transition pattern is as expected for convex 86 Sr vs. 87 Sr relationships with positive y -intercepts 1 . Indeed, the 86 Sr vs. 87 Sr relationships are statistically concave (negative y -intercept), convex (positive y -intercept) and linear (negative y -intercept) for shallow (early), transitional, and late (deep) times, respectively (Fig. 2 e). Although a combined linear function (upper right in Fig. 2 d) also fits the data and approximates the authors’ original conclusion, a constant exogenous source is unlikely. The deep data do not have a statistically significant curvature; thus, a linear function with a statistically significant negative y -intercept describes the relationship shown. Because the two polynomial equations (early and transition) have either negative or positive y -intercepts, the backgrounds (new starting points) also differ. Mixing diagrams (Fig. 2 d), curvature analysis of the 86 Sr vs. 87 Sr plots (whether the x 2 term for a best-fit polynomial function is significant) (Fig. 2 e) and an analysis of d 87 Sr/ d 86 Sr derivatives (Fig. 2 f) produce similar interpretations. Early time periods (shallow) suggest that 87 Sr: 86 Sr ratios are becoming less enriched. Transition time periods are becoming more enriched, and late time periods (deep) have an unchanging exogenous source. The 18 O: 16 O ratio time patterns for the ice core (Fig. 2 c) and stalagmite 87 Sr: 86 Sr ratio time patterns (Fig. 2 b) are very different. However, the stalagmite derivative (Fig. 2 f) and the unconfounded 18 O: 16 O ice core sample (Fig. 2 c) are in good agreement. Unfortunately, there are insufficient data (total elements not provided) to fully evaluate additional speleothem 18 O: 16 O data 16–18 presented by the original authors. Researchers collected excellent data. With a little additional effort, the conclusions of their study are much more powerful. The results and interpretations from their stalagmite study closely match well-accepted interpretations of an ice core sample from Greenland. 2.3. Summary of Results The data from twelve well -performed studies were selected as examples. A summary of these results is presented below. 2.3 .1. Spelothem Sr isotopes and climate change Results from Zhou et al. 15 Covered above Section 2.2 . 2.3.2. Sr isotopes and migration of ancient humans – details are provided in Supplementary Material 1 Montgomery and Evans 13 are correct. Resolving migration in human archaeological populations (Indigenous Machair dwellers and Immigrant Silica dwellers) with strontium mixing diagrams is required to define sources. Without a mixing diagram analysis, there would be little to report other than the mean 87 Sr: 86 Sr ratios being similar for both populations, with Immigrant Silicate dwellers having more variability. Researchers would wrongly conclude that both populations were exposed to similar 87 Sr: 86 Sr sources. 2.3.3. Details of the Sr isotopes and migration of an ancient cow are provided in Supplementary Material 1. Horstwood et al . 19 stated that the change in the direction of 87 Sr: 86 Sr with tooth age is opposite to what might be expected if the animal (a cow) moved from a high 87 Sr: 86 Sr source ratio to a lower one. Therefore, the authors suggest that the animal was likely to be slaughtered soon after moving to the low- 87 Sr: 86 Sr source before any tooth changes could occur. We agree with the conclusion of slaughter soon after arrival but suspect that there was no change in diet as the original authors suggested. The variation in 87 Sr: 86 Sr ratios from the cusp to the cervix that reflects early growth periods for the animal is entirely explained by differences in total Sr. There was likely no change in diet as the tooth developed. 2.3.4. Sr isotopes and migration of a prehistoric mammoth – Details in Supplementary Material 1 Isotopic researchers studying migration often assume that an observed isotopic ratio in a tissue reflects the source of a system, but this assumption is often not true. Sources derived from mammoth tooth data Kowalik et al. 20 , rather than observed ratios, should have been used to make matches with geographic areas. Furthermore, changing source patterns in an animal will be difficult when matching an unchanging source value for a specific location. Source evaluations for the time period of interest must be constant. This was not the case in the data evaluated. 2.3.5. Tracking Sr isotope changes in a watershed – Details in Supplementary Material 1 As theoretically demonstrated 1 , differences in the patterns of observed 87 Sr: 86 Sr ratios and sources in a water inlet Bailey et al. 21 can be explained by changes in total Sr. The time course variability is greater for sources than for observed ratios. This is expected since a large isotopic pool size buffers the observed ratio changes. 2.3.6. Sea spray as a possible Sr source in soils. – Details in Supplementary Material 1 Whipkey et al . 14 are likely correct that seawater could be a Sr source in this Hawaiian soil example. However, conventional techniques overestimate the magnitude of sea spray input. Contrary to what the authors propose, the portion of Sr derived from seawater is both small and does not markedly decrease among the deeper depths. 2.3.7. Nitrogen isotopes and trophic levels – details are provided in Supplementary Material 2 Moyo et al. 22 suggested that sparrows occupy a slightly greater trophic level (~ 0.7 delta 15 N unit difference) than marsh rats. However, when the five proposed alternate source methods were averaged, the sparrows and marsh rats were more different than the simple 15 N: 14 N ratios suggested. Sparrows have a 15 N: 14 N source ratio of 0.003759 vs. 0.003737 for marsh rats (an ~ 2.7 delta unit difference). Conventional trophic level assignment also overestimates the trophic level assigned to phytoplankton. 2.3.8. Nitrogen isotopes in a greenhouse tracer study – Details in Supplementary Material 2 Sandrock et al . 23 suggested that the significant differences in the 15 N: 14 N ratio and NDFF for E. alatus are caused by scaling effects rather than differences in physiology. These differences are indirectly caused by the smaller total N composition associated with a smaller plant size. One should not conclude that E. alatus differs in its physiology and is fundamentally more efficient than the other two species evaluated. 2.3.9. Evaluation of C isotopes on Mars as evidence for past life – Details in Supplementary Material 2 House et al . 24 proposed three explanations for depleted Martian 13 C values: biological methane production, photoreduction of atmospheric CO 2 , and cosmic dust. All three of these explanations are unconventional and differ from processes common on Earth. Although all sediments have samples with depleted observed 13 C: 12 C ratios, it is striking that data from the oldest sediments produce evidence for depleted 13 C sources where the three younger formations do not. As is the case with climate change, migration, trophic level, and tracer studies discussed above, sources rather than observed ratios are what should be evaluated in searches for past life on Mars. 2.3.10. Evaluation of C isotopes on Mars – Details in Supplementary Material 2 House et al . 24 also evaluated S isotopes for the same sediments on the basis of their evidence from past life studies conducted on the Martian samples described above. Total S confounds 34 S: 32 S evaluations of the same sediments. 2.3.11. Challenging N isotope fractionation in ocean sediments – Details are provided in Supplementary Material 3. Constant negative Rayleigh N fractionation factors suggesting preferences for 14 N, the common isotope, which was calculated for sediments in the Aegean Sea 25, are challenging. The significant ln total N vs delta 15 N slopes that define fractionation are driven by covarying total N. Issues are apparent for data with both linear and curvilinear 14 N vs. 15 N plots. Apparent fractionation can have alternate explanations. There may not be a constant microbial preference for lighter common isotopes defined by a fractionation factor. Rather, significant Rayleigh fractionation factors could be caused by relatively constant 15 N: 14 N loss ratios even though substrate isotopic ratios vary with depth. 2.3.12. Challenging C isotope fractionation in soils – Details in Supplementary Material 3 Constant negative Rayleigh C fractionation factors 26 suggesting preferences for 12 C, the common isotope, which was calculated for soils on the Himalayan Plateau, are challenging. The significant ln total C vs delta 13 C slopes that define fractionation are driven by covarying total C. Issues are apparent for the linear 12 C vs. 13 C plots that were evaluated. Apparent fractionation can have alternate explanations. There may not be a constant microbial preference for lighter common isotopes defined by a fractionation factor. Rather, significant Rayleigh fractionation factors could be caused by relatively constant 13 C: 12 C loss ratios even though substrate isotopic ratios vary with depth. 3. Discussion In summary, total Sr confounds 87 Sr: 86 Sr ratios. Specifically, source evaluations rather than analyses of ratio-based expressions allow one to better 1) utilize 87 Sr: 86 Sr ratios in migration patterns for ancient humans and animals, 2) associate speleothem 87 Sr: 86 Sr isotope data with known climate changes, 3) define the dynamics of 87 Sr: 86 Sr isotope data within a watershed and 4) define Sr sources in a soil. Three 15 N: 14 N, two 13 C: 12 C and one 34 S: 32 S studies were also re-evaluated. Total N confounds the interpretation of 15 N: 14 N studies on 1) isotopic fractionation in the Aegean Sea, 2) N tracers in potted plants and 3) trophic level studies in a marsh. Total C confounds 13 C: 12 C studies on 1) isotopic fractionation in soil and 2) an evaluation of whether past life existed in Martian sediments collected by the Curiosity rover. Total S confounds 34 S: 32 S studies in the same Martian sediments. In much of the ratio literature, nonzero y -intercepts in denominator vs. numerator plots mathematically explain why a ratio is related to denominator size 8,29 . However, nonzero y -intercepts are counterintuitive and conceptionally awkward. For example, a positive y -intercept for an 86 Sr vs. 87 Sr relationship could suggest that as total Sr (or the closely related 86 Sr) disappears, some 87 Sr still exists. Why would some 87 Sr still be present when total Sr is zero? Similarly, a negative y -intercept implies that a negative amount of 87 Sr exists as the total Sr approaches zero. How can there be a negative value for an isotope? This inconsistency is explained by the fact that nonzero y -intercepts have chemical meaning. For both linear and polynomial denominators vs. numerators, isotopic functions y -intercepts are related to backgrounds. Because they are not normally distributed, Isles 6 suggested that using raw (nontransformed) ratios to interpret elemental ratio data is inappropriate. He suggested that log-transformed or geometric means should be used instead and provides an example where nontransformed elemental N:P ratios provide misleading interpretations for very large datasets with a broad range of N:P ratios. Although we agree with Isles’s 6 concerns that ratio-based expressions are often misleading, these specific issues do not apply to isotopic ratios that vary over a narrow range. However, before evaluating the raw isotopic ratios and the slopes and y -intercepts of plots of their denominators and numerators, we first determined that using arithmetic means, log-transformed means, or geometric means of isotopic ratios led to the same interpretation of differences among times, depths, or treatments. We have not found any cases where Isles’s 6 transformation concerns apply to isotopic ratios (79Sr datasets, including examples here and in Supplementary Material S; data not shown but available on request). Arguments have also been presented where the use of log-transformed data has been challenged (the opposite of Isles’s 6 concerns expressed above). The great influence of scaling studies has attracted scrutiny as to the validity of using log-transformed data when evaluating scaling relationships 30–33 . Each of these publications has been challenged 34–37 . We agree with these challenges when the plotted values range over several orders of magnitude. However, there are still many advantages to using nontransformed data. We therefore advocate the use of nontransformed data to evaluate isotopic scaling. Nontransformed data are especially useful because determining whether denominator vs. numerator plots are linear is important. One wants to know if sources are constant or changing. A consistent curvature in a raw data plot can be verified with significant x 2 terms in the polynomial denominator vs. numerator analyses or nonlinear mixing diagrams. All the artificial perfectly linear functions in Fig. 1 a and c in our companion paper 1 that have nonzero y -intercepts can be approximated with log-log functions that have r 2 values greater than 0.999999 with slopes that significantly differ from 1.0 (data not shown). However, in all the cases, the implied curvatures indicated by the significant power functions fit truly linear functions, suggesting that changing sources do not exist. Slopes for nontransformed linear denominator vs. numerator plots define sources. The nonzero y -intercepts for linear or polynomial nontransformed denominator vs. numerator functions provide interpretive information regarding backgrounds. This being said, we see no harm in using either approach (as presented in this report) to glean more information. One should always evaluate traditional Keeling plots, other forms of mixing diagrams, or an analysis of the relationship between the ratio denominator and the numerator. Source evaluations inform researchers when isotopic ratios are confounded. Many datasets may not have ratio issues; thus, ratio-based assessments can still be useful. However, when total element data are available, finding confounded ratio examples is not difficult. One will never know if isotopic data are confounded unless they are evaluated. 4. Methods 4.1. Untangling Confounded Data: Five Approaches Isotopic ratios that are confounded by different denominator sizes will be misleading. There are many ways to compensate for the errors dictated by mathematical theory. The five procedures proposed here allow one to define the relationship between numerators and denominators more clearly: 1) Plotting the denominator vs. the numerator concentrations (both in umole/g units) will produce a slope that is equal to the source (m in the y = mx + b equation for a denominator vs. numerator plot) for linear functions. The denominator vs. numerator plot can be evaluated to determine whether a source is consistently increasing or decreasing by evaluating the significance of the x 2 term of best-fit polynomial equations. 2) Adjusting standard isotopic ratios (always calculated on a molar basis) to account for nonzero y -intercepts removes the background and total element effects that confound interpretation of sources. The value of a positive y -intercept for a denominator vs. numerator plot (in umolar units) can be subtracted from the numerator isotope values of a dataset before ‘new‘ ratios are recalculated with the original denominators. Similarly, the absolute value of a negative y -intercept can be added to the numerator isotope values of a dataset, after which a newly calculated ratio can be calculated from the original data. For polynomial denominator vs. numerator functions, a second y -intercept correction method uses a feature of polynomial equations where the ratios for a function with a zero intercept can be calculated from the terms of the original 86 S vs. 87 Sr relationships, regardless of whether the original function has a zero y -intercept 1 . For both linear and curvilinear functions, procedures should not be viewed as an extrapolation of a value beyond the range of points presented by the data. Instead, a y -intercept correction for a linear function can be viewed as indirectly and mathematically collecting points that define the constant derivative (source) for the range of points evaluated. This defines a source for each individual data point. An average of all y -intercept adjusted points approximates the slope of the original linear denominator vs. the numerator function. For nonlinear functions, adjusted y -intercepts do not produce a collection of points that define the derivative; rather, these points define the pattern of ratio changes as the denominator increases for a curvilinear function with a zero y -intercept. 3) The y -intercept defines the source (molar-based isotopic ratio) for traditional mixing diagram plots where 1/total element concentration (usually expressed as µg/g or ppm) is plotted vs. isotopic ratio (calculated on a molar basis) . This occurs because, at an infinite elemental concentration (1/element concentration approaches zero), the effect of the original background has been diluted, and the observed isotopic ratio approaches the source of material being added. 4) The y -intercept defines the source (molar-based isotopic ratio) as follows: 1/denominator (umole/g units) is used as the x -axis rather than the traditional 1/total element concentration. The modified mixing diagram is mathematically linked to the denominator vs. numerator function. If the x -axis is expressed in 1/umole/g units, the slope of a modified (1/denominator vs. isotopic ratio) mixing diagram will equal the original denominator vs. numerator y -intercept, and the mixing diagram y -intercept will equal the original denominator vs. numerator slope. With less than perfect real data, the relationships between the modified mixing diagram slope and the original denominator and the numerator y -intercept approximate one another but may not be exact matches. 5) Derivatives can be calculated from the denominator vs. numerator data. For a given denominator, the derivative represents the source (molar isotopic ratio) if both the numerator and denominator are calculated on a molar basis. A flat derivative vs. denominator concentration plot suggests that the sources are constant. A changing derivative suggests changing sources. There are many ways to calculate a numeric derivative. Here, we used a simple spine smoothing cubic option in SYSTAT TableCurve ® 2D software. The software can produce derivative values for a wide range of isotopic values in the original dataset. This information can be used to produce derivative values (interpolation for an estimation of source) for every observation. 4.2 Overview of Data Examples We have supplemented past theoretical explanations, artificial data examples, simulations and companion paper 1 with detailed analyses of Sr examples from both a modified dataset (Section 2.1 ) and the cave data (Section 2.2 ) discussed above. Additional 87 Sr: 86 Sr data from the following publications are presented: ( 1 ) Montgomery and Evans 13 , ( 2 ) Horstwood et al . 19 , ( 3 ) Kowalik et al . 20, ( 4 ) Zhou et al . 15 and ( 5 ) Bailey et al . 21 Publications ( 1 )-( 5 ) refer to studies on Sr sources in human teeth, migration possibilities for an ancient cow, migration possibilities for a European mammoth, a speleothem, and water from a river inlet, respectively. Another dataset, from Whipkey et al . 14, that addresses Sr sources in soil is presented in Supplementary Material S2. These are the original data that were modified to create the first artificial example (Section 2.1 , Untangling confounded sources of a constant source artificial example). Data re-evaluations involving isotopic analyses of N, C and S from the following publications were also conducted: ( 1 ) Moyo et al . 22 , ( 2 ) Sandrock et al . 23 , and ( 3 ) House et al . 24 Publication ( 1 ) addresses C and N isotope evaluations to evaluate trophic levels (only nitrogen isotopes were used in our re-evaluation); Publication ( 2 ) covers a 15 N-depleted tracer study on ornamental plants grown in pots; and Publication ( 3 ) involves an isotopic analysis of 13 C and 34 S on samples collected by the Curiosity rover on Mars. Data re-evaluations involving N 25 and C 26 isotopes are presented. Möbius et al . 25 evaluated N isotopes in Aegean Sea sediments. Wang et al . 26 conducted a soil C isotope study on the Tibetan Plateau. Both re-evaluations challenge the existing concepts of isotopic fractionation. As stated above, the details of all the dataset re-evaluations are presented in Supplementary Materials S1-S3. 4.3 Methods for Initial Artificial (Modified Data) Linear Denominator vs. Numerator Example Most 87 Sr: 86 Sr ratios in the literature are recorded up to the 5th decimal, and percentages of total 86 Sr and 87 Sr are reported to the 4th decimal in accordance with the National Institute of Standards and Technology Certificate of Analyses for the SRM 987 standard. All of the authors’ work with Sr isotopes was re-evaluated here, and in Supplementary Material 2, we used this standard. The standard values are 0.71039 ± 0.00013, 7.0015 ± 0.0026, and 9.8566 ± 0.0034 for the 87 Sr: 86 Sr ratio, percent 87 Sr and percent 86 Sr, respectively. Standard errors for the SRM 987 sample listed above are relatively high, but since mass spectrometers measure relative differences far more accurately than absolute differences reporting to the 5th decimal place when a reference sample is used is appropriate. Producing 86 Sr vs. 87 Sr plots for real datasets requires some assumptions because only the total isotope concentration and 87 Sr: 86 Sr ratios are known. Thus, not all isotopic species are accounted for. For the modified data, the portion for the sum of 86 Sr and 87 Sr was forced to remain constant and was equal to the standard values of 9.8566% + 7.0015%. This assumption was more fully evaluated for the real datasets described below. When total Sr concentrations are presented in the literature, they are usually expressed in ppm or µg/g units. These total concentration units can be converted to µm/ml or µm/g by assuming that the molecular weight, including all the Sr mass isotopes, is 87.62. The combined total for both 87 Sr and 86 Sr (in µmol/g units) can then be calculated (total * 0.168581). The values for both 87 Sr and 86 Sr (in µmol/g units) can then be calculated (total * 0.1685811). The 86 Sr value = combined 87 Sr and 86 Sr (in µmol/g units/ 87 SR: 86 Sr ratio + 1). The 87 Sr value = the combined total of 87 Sr and 86 Sr – 86Sr. In all the cases, the sums of the 87Sr + 86Sr values were checked to ensure that the value was 0.168581 * total Sr in µm. The 87 Sr: 86 Sr ratios (recalculated from the derived umolar/g values of 86 Sr and 87 Sr) matched the original published 87 Sr: 86 Sr ratios. Once the 87 Sr and 86 Sr concentrations are known, one can determine the statistical significance of the x 2 terms in the polynomial best-fit 86 Sr vs. 87 Sr equations and whether the descriptive denominator vs. numerator functions are linear, convex ( 87 Sr increases with increasing 86 Sr), or concave ( 87 Sr decreases with increasing 86 Sr). Spline-smoothed derivatives can also be determined (with TableCurve® 2D Sigma software). It has been our experience that spline smoothed cubic derivatives are not always helpful because some datasets have considerable variation. However, in other cases, these derivatives are extremely useful in interpreting datasets and often closely agree with mathematical derivatives derived from a polynomial best-fit denominator vs. numerator equation. Knowing whether mixing lines are linear also provides insight into whether the original denominator vs. numerator plot has subtle curvature. Mathematically, a linear denominator vs. numerator function must produce a linear mixing diagram. 4.4 Methods for All Real Data Examples Real 87 Sr: 86 Sr datasets were analyzed as described above. Changes in portions of specific isotopes in natural systems are extremely small, but it is important to determine whether slight differences in assumptions will alter the results and which assumptions are appropriate. The assumption that the sum of 86 Sr and 87 Sr is equal to the standard value of 9.8566% + 7.0015% was validated by comparing the results of two other assumptions: 1) the portion of 86 Sr remains constant and is equal to the standard value of 9.8566%, and 2) the portion of 87 Sr remains constant and is equal to the standard value of 7.0015%. The 9.8566% + 1.0015% assumption produces intermediate 86 SR vs. 87 Sr slope and y -intercept values that are only slightly different than the results for assumptions 1) and 2) above. Therefore, the constant 87Sr n + 86Sr assumption was used thereafter. The evaluation of all three assumptions in Supplemental Material 1, which addresses the migration of ancient humans 13, suggests that the 86 Sr vs. 87 Sr slope, calculated with a constant 87 Sr plus 86 Sr assumption, produces an appropriate estimate of sources. The theoretical, simulated, and real data for Sr isotope evaluations (with the previously stated assumption) all suggest problems interpreting Sr isotope ratios. Without this Sr assumption, one will be forced to conclude that although cases where Sr isotope ratios are confounded by differences in total elements are rampant, there is no way to fully re-evaluate the Sr literature, calculate 86 Sr vs. 87 Sr slopes and derivatives, and determine whether sources are constant or statistically changing. However, although rarely used, traditional mixing diagram approaches using raw data can still be applied to define isotopic Sr sources if linear mixing lines are produced. In other analyses (Supplementary Materials 2 and 3), non-Sr isotope re-evaluation examples are presented. The data reanalyses were performed as explained in the initial Sr isotope examples explained above. However, no assumptions (required when an element has more than two isotopic species) were used when calculating denominator or numerator values for N and C studies. Nitrogen and C have only two major mass isotopes. Sulfur has more than two other isotope species. However, denominator vs. numerator plots were not constructed for S isotopes because traditional mixing diagrams easily define sources. Slopes and derivatives reflect constant and changing sources, respectively. Changes in sources for different layers or samples over time can be determined for re-evaluated datasets by matching derivatives calculated from 86 Sr vs. 87 Sr relationships to specific values of 86 Sr or total Sr. All new information was interpreted to determine if the original authors’ conclusions were supported. Important details for all the isotopic data reanalyses are presented in the supplementary material (Supplemental Material 1, Data re-evaluation for SR isotopes; Supplemental Material 2, Data re-evaluations for N, C and S isotopes; and Supplemental Material 3, Rethinking fractionation favoring the common isotope. Declarations Competing Interests The author(s) declare no competing interests. Author Contribution TLR is the team leader and was responsible for developing concepts over many years. CPN and CN spent considerable time investigating ratio-related issues that included isotope evaluations during their Master of Science work under TLR that in part led to this publication. CPN and CN also made substantial editing contributions. KM was the major editing resource for the team and responsible for improving the initial rough draft of the manuscript produced by TLR. FC and DL made editing contributions that were focused on isotopic scaling issues directed at readers that had limited exposure to both isotopic research and ratio-related scaling issues. Acknowledgement Sincere thanks and love of TLR’s life Mary Ann Righetti (BS,MLS,JD). She has been a sounding board for the many years of concept development. Also appreciated are a group of undergraduate students that edited this manuscript. We reasoned that for a publication this controversial to be accepted, every sentence in the text and figure legends should be understandable by a well-trained undergraduate. Students spent countless hours improving the manuscript. Their names are listed in supplemental material. Data Availability All data generated or analysed during this study are included in this published article [and its supplementary information files]. References Moots, K. Nguyen, C. P Nguyen C. Camacho, F., Lindstrom, D., and Righetti, T, L.. A mathematical explanation for why ratio-based isotopic analyses are commonly misleading: Theory. Scientific Reports this issue (2024). Atchley, W. R., Gaskins, C. T. & Anderson, D. Statistical properties of ratios. I. Empirical results. Syst. Biol. 25, 137–148 (1976). Jackson, D. A., Harvey, H. H. & Somers, K. M. Ratios in aquatic sciences: statistical shortcomings with mean depth and the morphoedaphic index. Can. J. Fish. Aquat. Sci. 47, 1788–1795 (1990). Jasieński, M. & Bazzaz, F. A. The fallacy of ratios and the testability of models in biology. Oikos 84, 321–326 (1999). Kratochvíl, L. & Flegr, J. Differences in the 2nd to 4th digit length ratio in humans reflect shifts along the common allometric line. Biol. Lett. 5, 643–646 (2009). Packard, G. C. & Boardman, T. J. The misuse of ratios, indices, and percentages in ecophysiological research. Physiol. Zool. 61, 1–9 (1988). Isles, P. D. F. The misuse of ratios in ecological stoichiometry. Ecology 101, e03153 (2020). Righetti, T. 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Determination of Sr isotopes in calcium phosphates using laser ablation inductively coupled plasma–mass spectrometry and their application to archaeological tooth enamel. Geochim. Cosmochim. Acta 72, 5659–5674 (2008). Kowalik, N. et al. Revealing seasonal woolly mammoth migration with spatially resolved trace element, Sr and O isotopic records of molar enamel. Quat. Sci. Rev. 306, 108036 (2023). Bailey, S. W., Hornbeck, J. W., Driscoll, C. T. & Gaudette, H. E. Calcium inputs and transport in a base-poor forest ecosystem as interpreted by Sr isotopes. Water Resour. Res. 32, 707–719 (1996). Moyo, S. et al. Stable isotope analyses identify trophic niche partitioning between sympatric terrestrial vertebrates in coastal saltmarshes with differing oiling histories. PeerJ 9, e11392 (2021). Sandrock, D. R., Righetti, T. L. & Azarenko, A. N. Isotopic and nonisotopic estimation of nitrogen uptake efficiency in container-grown woody ornamentals. HortScience 40, 665–669 (2005). House, C. H. et al. Depleted carbon isotope compositions observed at gale crater, mars. Proc. Natl. Acad. Sci. 119, e2115651119 (2022). Möbius, J. Isotope fractionation during nitrogen remineralization (ammonification): implications for nitrogen isotope biogeochemistry. Geochim. Cosmochim. Acta 105, 422–432 (2013). Wang, G., Jia, Y. & Li, W. Effects of environmental and biotic factors on carbon isotopic fractionation during decomposition of soil organic matter. Sci. Rep. 5, 11043 (2015). Grootes, P. M., Stuiver, M., White, J. W. C., Johnsen, S. & Jouzel, J. Comparison of oxygen isotope records from the GISP2 and GRIP Greenland ice cores. Nature 366, 552–554 (1993). Stuiver, M., Braziunas, T. F. & Grootes, P. M. The GISP2 δ18O climate record of the past 16,500 years and the role of the sun, ocean, and volcanoes. Quat. Res. 44, 341–354 (1995). Tanner, J. M. Fallacy of per-weight and per-surface area standards, and their relation to spurious correlation. J. Appl. Physiol. 2, 1–15 (1949). Packard, G. C. On the use of logarithmic transformations in allometric analyses. J. Theor. Biol. 257, 515–518 (2009). Packard, G. C. Is logarithmic transformation necessary in allometry? Biol. J. Linn. Soc. 109, 476–486 (2013). Packard, G. C. Misconceptions about logarithmic transformation and the traditional allometric method. Zoology (Jena) 123, 115–120 (2017). Packard, G. C. & Boardman, T. J. Model selection and logarithmic transformation in allometric analysis. Physiol. Biochem. Zool. 81, 496–507 (2008). Glazier, D. S. Log-transformation is useful for examining proportional relationships in allometric scaling. J. Theor. Biol. 334, 200–203 (2013). Kerkhoff, A. J. & Enquist, B. J. Multiplicative by nature: why logarithmic transformation is necessary in allometry. J. Theor. Biol. 257, 519–521 (2009). Mascaro, J., Litton, C. M., Hughes, R. F., Uowolo, A. & Schnitzer, S. A. Is logarithmic transformation necessary in allometry? Ten, one-hundred, one-thousand-times yes. Biol. J. Linn. Soc. 111, 230–233 (2014). Pélabon, C., Tidière, M., Lemaître, J.-F. & Gaillard, J.-M. Modelling allometry: statistical and biological considerations – a reply to Packard. Biol. J. Linn. Soc. 125, 664–671 (2018). Additional Declarations No competing interests reported. Supplementary Files 4ConfoundedSupplemental1tlrmar2.docx 5ConfoundedSupplemental2tlrmar2.docx 6ConfoundedSupplemental3tlrmar2.docx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4086527","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":297043082,"identity":"c0564e2c-159f-4cdb-b1ee-d2d0744a5e5d","order_by":0,"name":"Kate Moots","email":"","orcid":"","institution":"University of Guam","correspondingAuthor":false,"prefix":"","firstName":"Kate","middleName":"","lastName":"Moots","suffix":""},{"id":297043087,"identity":"e2ce9fba-4cf9-49d6-8b44-4d8edc564c61","order_by":1,"name":"Christina P. Nguyen","email":"","orcid":"","institution":"University of Guam","correspondingAuthor":false,"prefix":"","firstName":"Christina","middleName":"P.","lastName":"Nguyen","suffix":""},{"id":297043094,"identity":"2202a8ba-dddb-48f3-ac9b-0ac5b0283369","order_by":2,"name":"Catherine Nguyen","email":"","orcid":"","institution":"University of Guam","correspondingAuthor":false,"prefix":"","firstName":"Catherine","middleName":"","lastName":"Nguyen","suffix":""},{"id":297043101,"identity":"4275ed0c-23c2-4606-b844-ba588cad9e1b","order_by":3,"name":"Frank Camacho","email":"","orcid":"","institution":"University of Guam","correspondingAuthor":false,"prefix":"","firstName":"Frank","middleName":"","lastName":"Camacho","suffix":""},{"id":297043106,"identity":"d7f083a9-80af-49ee-a5f1-8dab1212167a","order_by":4,"name":"Dan Lindstrom","email":"","orcid":"","institution":"University of Guam","correspondingAuthor":false,"prefix":"","firstName":"Dan","middleName":"","lastName":"Lindstrom","suffix":""},{"id":297043110,"identity":"d6c4f48f-65a1-4e9f-8ae9-b49690d7edbf","order_by":5,"name":"Timothy L. Righetti","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA9klEQVRIie3OPYoCMRTA8TcEZpqIrYPOnOFJQBFEzzII1oKNlUQWYiNaewvtLCMpbDzEDnsBBxQVBI0fuIJGLQXzJ6R45McLgM32oUmH65sQCYD/09PwUfRCMpS4EQfE9wicCNA8v11jJFVIxdPVpNyuenT912rswqK28RLKwUiatniognldfyw17swRWYlTxnJQZ2bigvKFOhOOGI0kLWQzoKIXZK8JjW/J/imZJkIeiXMl/gKkmSi9xRE1v6fc/JAjY6jcJgOssaGBeF1Bkq2opL2B+k34Lgxx9jOON61K0DcQIPrQ+wkanl9yNq8nNpvN9s0dAMLLVWNZNfQjAAAAAElFTkSuQmCC","orcid":"","institution":"University of Guam","correspondingAuthor":true,"prefix":"","firstName":"Timothy","middleName":"L.","lastName":"Righetti","suffix":""}],"badges":[],"createdAt":"2024-03-12 16:56:24","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4086527/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4086527/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":55763818,"identity":"22ab3535-8aa0-4cef-bfd3-69d6abc5ed42","added_by":"auto","created_at":"2024-05-02 19:44:01","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":241831,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSr isotope relationships for a constant source system using modified real data evaluating sea spray as a possible soil Sr source\u003c/strong\u003e. \u003cstrong\u003ea\u003c/strong\u003e. Relationships between the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio and total Sr with soil depth from a real dataset\u003csup\u003e14\u003c/sup\u003e that has been modified. This dataset demonstrates how less than perfectly linear relationships with levels of variability similar to what is encountered in real datasets can produce \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios that are driven by changes in total Sr (or the highly correlated \u003csup\u003e86\u003c/sup\u003eSr). The data were further modified to create subtle \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr concave curvatures (\u003cem\u003ex\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e term \u003cem\u003ep \u003c/em\u003e= 0.14), which were not statistically significant. The original data had significant concave curvature if the deepest point was excluded from the analysis. This characteristic was removed, and both the entire modified dataset and the top four depths were described by similar equations (data not shown). The values for total Sr and \u003csup\u003e86\u003c/sup\u003eSr have also been modified to always increase with depth (unlike the original data). Furthermore, the range of \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios has been modified to create a larger range between two logical but not necessarially accurate endmember sources (seawater and basalt). The basalt soil parent material was suggested by the original authors. The numbers adjacent to the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio points indicate the percentage of Sr at a given depth that originated from seawater (using standard formulas); \u003cstrong\u003eb.\u003c/strong\u003e The relationship between \u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e87\u003c/sup\u003eSr from a real dataset\u003csup\u003e14\u003c/sup\u003e that has been modified to create subtle concave curvature (\u003cem\u003ex\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e term \u003cem\u003ep \u003c/em\u003e= 0.14) that is neither visible nor statistically significant. The details of the dataset construction are described in Fig. 1\u003cstrong\u003ea\u003c/strong\u003e. The function is linear, and the \u003cem\u003ey\u003c/em\u003e-intercept, although very small, is statistically significant (\u003cem\u003ep \u003c/em\u003e= 0.00012). \u003cstrong\u003ec.\u003c/strong\u003e Relationships between the \u003csup\u003e86\u003c/sup\u003eSr (umoles/g) and \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios from a real dataset\u003csup\u003e14\u003c/sup\u003e that has been modified. The predicted values are derived from the linear \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr equation presented in Fig. 1\u003cstrong\u003eb\u003c/strong\u003e. The predicted values are an almost exact match for values from the best-fit \u003cem\u003ey\u003c/em\u003e = (1/\u003cem\u003ex\u003c/em\u003e) + b equation (values not shown); \u003cstrong\u003ed\u003c/strong\u003e. The 1/denominator (1/\u003csup\u003e86\u003c/sup\u003eSr) form of a mixing diagram where 1/\u003csup\u003e86\u003c/sup\u003eSr is plotted against observed and \u003cem\u003ey\u003c/em\u003e-intercept adjusted ratios. Additionally, a slightly changing spline smoothed cubic derivative (small gray points) and the specific \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr derivative values (large gray points) associated with known \u003csup\u003e86\u003c/sup\u003eSr values for the five depths are presented. The slope of the raw data mixing line (0.00000007105) approximates the \u003cem\u003ey\u003c/em\u003e-intercept (0.000000071786), as shown in Supplementary Fig. S1\u003cstrong\u003eb\u003c/strong\u003e. The \u003cem\u003ey\u003c/em\u003e-intercept of this mixing line (0.69859) closely matches the slope (0.69853) of the original \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr plot in Fig. 1\u003cstrong\u003eb\u003c/strong\u003e. A mixing diagram constructed from \u003cem\u003ey\u003c/em\u003e-intercept adjusted points is also shown. The \u003cem\u003ey\u003c/em\u003e-intercepts were adjusted by subtracting the linear positive \u003cem\u003ey\u003c/em\u003e-intercept shown in Fig. 1\u003cstrong\u003eb\u003c/strong\u003e from the original \u003csup\u003e87\u003c/sup\u003eSr values in the dataset and creating a new \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio. Modified data after Whipkey \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e14\u003c/sup\u003e; \u003cstrong\u003ee\u003c/strong\u003e. Five different source estimates (described in Section 1) plotted vs. depth. The five sources are the \u003cem\u003ey\u003c/em\u003e-intercept corrected ratios, spline smoothed cubic derivative, \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr slope, and both traditional 1/total Sr (ug/g units) and 1/\u003csup\u003e86\u003c/sup\u003eSr (umole/g units) mixing line \u003cem\u003ey\u003c/em\u003e-intercepts. Modified data after Whipkey \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e14\u003c/sup\u003e; \u003cstrong\u003ef.\u003c/strong\u003e Five different source estimates (all shown in Fig. 1\u003cstrong\u003ee\u003c/strong\u003e) plotted vs. depth along with the original \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr observed ratios. The five sources are the \u003cem\u003ey\u003c/em\u003e-intercept corrected ratios, spline smoothed cubic derivative, \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr slope, and both traditional 1/total Sr (ug/g units) and 1/\u003csup\u003e86\u003c/sup\u003eSr (umole/g units) mixing line \u003cem\u003ey\u003c/em\u003e-intercepts. Modified data after Whipkey \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e14\u003c/sup\u003e.\u003c/p\u003e\n\u003cp\u003eAll panels are available as individual files in *.pdf format.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4086527/v1/d8603b49f6e31bef9db099c4.png"},{"id":55763817,"identity":"84df9d62-c077-41b8-bb79-01f0060b55b9","added_by":"auto","created_at":"2024-05-02 19:44:01","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":215046,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSr isotope relationships for a changing source system using data from a cave stalagmite evaluating possible connections between the cave data and climate change\u003c/strong\u003e. \u003cstrong\u003ea.\u003c/strong\u003e Relationships between total Sr in stalagmite layers from Songjia Cave, Northeast Sichuan Province, Central China, where total Sr significantly varied during the period between 20,000 and 10,000 years before the present. b. Relationships between depth and the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio for a stalagmite from Songjia Cave, Northeast Sichuan Province, Central China, for the period between 20,000 and 10,000 years before the present. This figure matches 15 from Zhou \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e15\u003c/sup\u003e but has been rotated 90 degrees. However, unlike in the previously published figure, different depths with possibly different sources are shown.; \u003cstrong\u003ec.\u003c/strong\u003e Temporal patterns of \u003csup\u003e18\u003c/sup\u003eO\u003csup\u003e16\u003c/sup\u003eO ratios (delta \u003csup\u003e18\u003c/sup\u003eO) in Greenland ice cores\u003csup\u003e27,28\u003c/sup\u003e replotted by Zhou \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e15\u003c/sup\u003e and rotated 90 degrees here). The arrows in the figure refer to the early (blue), transition (green) and late (orange) time periods denoted in Fig. 2\u003cstrong\u003ea\u003c/strong\u003e and \u003cstrong\u003eb\u003c/strong\u003e. Long gray bars indicate two known cold periods. The \u003csup\u003e18\u003c/sup\u003eO:\u003csup\u003e16\u003c/sup\u003eO ratios clearly decrease, increase, and then remain relatively stable for time periods corresponding to shallow, transitional and deep stalagmite depths, respectively, as shown in Fig. 2\u003cstrong\u003ea\u003c/strong\u003e and \u003cstrong\u003eb\u003c/strong\u003e. Data from Zhou \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e15\u003c/sup\u003e; \u003cstrong\u003ed.\u003c/strong\u003e Mixing diagrams (1/\u003csup\u003e86\u003c/sup\u003eSr form) for shallow (blue), transitional (green) and deep (orange) portions of a stalagmite from Songjia Cave, Northeast Sichuan Province, Central China, for points that could be clearly assigned to different depths. A combined mixing (\u003cem\u003ey \u003c/em\u003e= -0.0089\u003cem\u003ex\u003c/em\u003e + 0.7105) line is also presented (top equation), which closely aligns with the authors’ (their Fig. 9\u003csup\u003e15\u003c/sup\u003e) 1/total Sr source prediction (0.7109 vs. 0.7105). e. Relationships between \u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e87\u003c/sup\u003eSr for a stalagmite from Songjia Cave, Northeast Sichuan Province, Central China. A concave polynomial (\u003cem\u003ex\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e term statistically significant) with a statistically significant negative \u003cem\u003ey\u003c/em\u003e-intercept fit the shallow data. A convex polynomial (\u003cem\u003ex\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e term statistically significant) with a statistically significant positive \u003cem\u003ey\u003c/em\u003e-intercept fit the transition data. A linear function with a negative \u003cem\u003ey\u003c/em\u003e-intercept fits the deep data. Data after Zhou \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e15\u003c/sup\u003e; \u003cstrong\u003ef.\u003c/strong\u003e Derivatives calculated from the relationship between \u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e87\u003c/sup\u003eS for shallow (blue), transitional (green) and deep (orange) portions of a speleothem from Songjia Cave, Northeast Sichuan Province, Central China. By matching spline smoothed derivatives calculated from \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr relationships to specific values of \u003csup\u003e86\u003c/sup\u003eSr or total Sr values that correspond to specific layers or time periods associated with depth, changes in sources for different layers or over time can be determined. Data after Zhou \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e15\u003c/sup\u003e\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4086527/v1/90fe414b388735a2691ab0bd.png"},{"id":56909833,"identity":"bce1065a-871e-44fe-ab24-4ca2b10907f6","added_by":"auto","created_at":"2024-05-22 04:39:37","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1561206,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4086527/v1/9ed89350-8506-48a7-ab2d-5a5a29a78ba8.pdf"},{"id":55763822,"identity":"8fae453b-2029-45a2-b7fa-212ad603486e","added_by":"auto","created_at":"2024-05-02 19:44:01","extension":"docx","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":490424,"visible":true,"origin":"","legend":"","description":"","filename":"4ConfoundedSupplemental1tlrmar2.docx","url":"https://assets-eu.researchsquare.com/files/rs-4086527/v1/d2b0b0186e3d50cf9682a5e5.docx"},{"id":55763819,"identity":"25fbfae0-896f-46a4-95c5-0a91ad023bab","added_by":"auto","created_at":"2024-05-02 19:44:01","extension":"docx","order_by":3,"title":"","display":"","copyAsset":false,"role":"supplement","size":321579,"visible":true,"origin":"","legend":"","description":"","filename":"5ConfoundedSupplemental2tlrmar2.docx","url":"https://assets-eu.researchsquare.com/files/rs-4086527/v1/0c7cc7dd22f7fce9bdfa8891.docx"},{"id":55763821,"identity":"f2266117-25b2-418c-819f-4d4f449500ec","added_by":"auto","created_at":"2024-05-02 19:44:01","extension":"docx","order_by":4,"title":"","display":"","copyAsset":false,"role":"supplement","size":361753,"visible":true,"origin":"","legend":"","description":"","filename":"6ConfoundedSupplemental3tlrmar2.docx","url":"https://assets-eu.researchsquare.com/files/rs-4086527/v1/1c38340122b1b8f798ddae39.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Mathematical Explanation for Why Ratio-Based Isotopic Analyses are Commonly Misleading: Dealing with Confounded Isotopic Ratios","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eIsotopic researchers often assume that observed isotopic ratios reflect the source (isotopic ratio of material gained or lost) of a system. However, mathematical principles suggest that observed ratios in isotopic samples almost always differ from those of sources\u003csup\u003e1\u003c/sup\u003e. Observed ratios when analyzed without evaluating sources can be misleading\u003csup\u003e1\u003c/sup\u003e. Issues related to ratio use in scientific research\u003csup\u003e2\u0026ndash;7\u003c/sup\u003e have been ignored for many decades\u003csup\u003e4,8\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eWe advocate using isotopic denominator vs. numerator plots to define isotopic sources. Keeling plots\u003csup\u003e9\u003c/sup\u003e and mixing diagrams\u003csup\u003e10\u0026ndash;13\u003c/sup\u003e have been accepted approaches for identifying isotopic sources for more than 60 years. However, purely mathematical concepts reveal that Keeling plots and isotopic mixing diagrams are indirect ways to calculate the slope and relative \u003cem\u003ey\u003c/em\u003e-intercepts of linear denominators vs. numerator isotope plots\u003csup\u003e1\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eThe only way in which a zero \u003cem\u003ey\u003c/em\u003e-intercept for a linear denominator vs. numerator isotope function can mathematically occur is when sources and backgrounds are similar, which produces flat mixing diagrams and constant isotopic ratios\u003csup\u003e1\u003c/sup\u003e. Therefore, nonzero \u003cem\u003ey\u003c/em\u003e-intercepts for linear denominator vs. numerator concentration plots are common, mathematically dictating that isotopic ratios are often related to their denominator or total elemental concentration\u003csup\u003e1\u003c/sup\u003e. Nonlinear relationships between denominator vs. numerator isotope concentrations affect how isotopic ratios change with increasing total element concentration in ways that are intuitively difficult to predict\u003csup\u003e1\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eTotal element concentrations, necessary for source analyses, are present in less than half of the isotopic publications\u003csup\u003e1\u003c/sup\u003e. Without evaluating the total elemental content, the relative background and source ratios cannot be determined. Even when total element data are available, researchers rarely conduct source analyses\u003csup\u003e1\u003c/sup\u003e. This approach is unfortunate because determining sources solves most interpretive issues. Derivatives (\u003cem\u003ed\u003c/em\u003enumerator/\u003cem\u003ed\u003c/em\u003edenominator) define changing sources. Nonzero \u003cem\u003ey\u003c/em\u003e-intercepts quantify backgrounds for linear and curvilinear polynomial functions\u003csup\u003e1\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eThe portion of an element derived from an exogenous source is traditionally calculated from isotopic ratios\u003csup\u003e1\u003c/sup\u003e. Isotopic preferences (fractionation or discrimination factors) are also derived from isotopic ratios. Therefore, both of these isotopic estimates are subject to the same confounding scaling effects as the original ratios. A paradigm shift from the use of simple observed isotopic ratios to procedures that directly identify source, background and total isotope effects on an isotopic ratio is necessary.\u003c/p\u003e"},{"header":"2. Results: Dealing with Confounded Isotopic Ratios","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1. Untangling Confounded Data: A Sample Analysis of a Constant Source Artificial Example\u003c/h2\u003e \u003cp\u003eSoil nutrients are generally thought to be derived primarily from rock weathering. This first example was calculated by modifying data from a study\u003csup\u003e14\u003c/sup\u003e in which Sr isotope data were collected at several depths on 30,000-year-old Pahala Ash deposits 50 m from the coast at South Point, Hawaii. The numbers adjacent to the hypothetical \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio points in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ea for different soil depths are calculated as percentages (using standard formulas) of the Sr at a given depth that originated from seawater spray added to the soil surface. Similar portions can be estimated by the relative position of a point between two possible sources (basalt and sea spray).\u003c/p\u003e \u003cp\u003eA typical interpretation based on \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios alone would suggest that seawater is an important source of Sr and that this influence decreases with depth. However, standard formulas used to calculate the portion of total elements derived from a suspected source do not account for the fact that the isotopic ratios used in these formulas are often confounded by nonzero \u003cem\u003ey\u003c/em\u003e-intercepts, which make isotopic ratios dependent on total element concentrations. It is possible that the Sr additions to the soil in this artificial example could be due to a blending of both seawater (0.7091) and basalt (0.7035) \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr sources, with the relative amounts of input from the two sources varying with depth. The value for the basalt source stated above and shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ea is for the basalt soil parent material suggested by the original authors.\u003c/p\u003e \u003cp\u003eThe original data can be interpreted differently depending on whether constant source or changing source analyses are conducted. To simplify this first data evaluation, the changing source explanation is much less likely for the modified data because a constant source (linear \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr relationship) adequately explains the results. In addition to demonstrating an alternative to ratio-based interpretations by providing examples of the five proposed ways of dealing with confounded ratios (Section \u003cspan refid=\"Sec18\" class=\"InternalRef\"\u003e4.1\u003c/span\u003e), this modified dataset also reveals how mathematical principles can produce surprising results. Even if the observed isotopic ratios fall within the range defined by suspected sources (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ea), the true source can be very different than either logically suspected source. A full analysis of the original unmodified data is included in the Supplementary Material, Section \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eSurprisingly, although total Sr and total Ca were measured, these data are not presented in Whipkey \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e14\u003c/sup\u003e Total Sr values were indirectly estimated before being modified for this example. The data were derived by first calculating the total Ca concentrations from the Na concentrations and molar Ca:Na ratios provided in the authors\u0026rsquo; original publication. Next, total Sr was calculated from the derived Ca concentrations and molar Sr:Ca ratios. Once the total Sr concentration was known, the umole/g values for \u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e87\u003c/sup\u003eSr could be calculated from the given \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr molar ratios using the assumption that the combined total portion of \u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e87\u003c/sup\u003eSr was 9.8566% + 7.0015. These two values are the respective portions of the total values for \u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e87\u003c/sup\u003eSr for the commonly used NBS SRM 987 Strontium Carbonate reference standard (used in all six Sr studies that were re-evaluated). Although, strictly speaking, concentration units are reserved for variables with units of volume, we use the broader definition commonly used in soils, sediments, and plant tissues (for example, \u0026micro;g/g or \u0026micro;mol/g).\u003c/p\u003e \u003cp\u003eThis modified dataset demonstrates how less than perfect linear relationships with levels of variability similar to what is encountered in real data can produce \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios that are driven by changes in total Sr (or the highly correlated \u003csup\u003e86\u003c/sup\u003eSr). This dataset also serves as an introduction to the fact that the denominator vs. numerator r\u003csup\u003e2\u003c/sup\u003e values presented in isotopic studies are spectacularly high. The data were altered to create a subtle concave curvature (polynomial \u003cem\u003ex\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e term \u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.14), which was not statistically significant. The values for total Sr and \u003csup\u003e86\u003c/sup\u003eSr have also been modified to always increase with depth (unlike the original data). Furthermore, the original range of observed \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios has been expanded to create a larger range between two logical but not necessarially accurate endmember sources (seawater and basalt).\u003c/p\u003e \u003cp\u003eThe relationship between \u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e87\u003c/sup\u003eSr for the modified data presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ea is presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003eb. The function is linear, and the \u003cem\u003ey\u003c/em\u003e-intercept, although very small, is statistically significant (\u003cem\u003ep\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.00012). It is difficult to determine how a small \u003cem\u003ey\u003c/em\u003e-intercept (approximately 0.485% of the \u003cem\u003ey\u003c/em\u003e value range) for a linear function could have interpretive consequences. However, mathematically, the ratio associated with a linear \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr function with a nonzero \u003cem\u003ey\u003c/em\u003e-intercept must change with increasing denominator size (\u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;m\u003cem\u003ex\u003c/em\u003e\u0026thinsp;+\u0026thinsp;b dictates \u003cem\u003ey\u003c/em\u003e/\u003cem\u003ex\u003c/em\u003e = (1/\u003cem\u003ex\u003c/em\u003e) b\u0026thinsp;+\u0026thinsp;m). This is apparent in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ec, where the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio decreases with increasing \u003csup\u003e86\u003c/sup\u003eSr, as predicted from the original \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr linear function with a positive \u003cem\u003ey\u003c/em\u003e-intercept shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003eb.\u003c/p\u003e \u003cp\u003eThe r\u003csup\u003e2\u003c/sup\u003e values for both linear and polynomial (\u003cem\u003ex\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e term not significant) best-fit equations are very high (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ea) but in the range of what one often encounters in real data\u003csup\u003e1\u003c/sup\u003e (see Supplementary Materials 1\u0026ndash;3 in reference 1). Differences in total Sr or strongly correlated \u003csup\u003e86\u003c/sup\u003eSr, both of which change with depth, drove the decrease in \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios rather than changes in exogenous sources (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ec). The \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr linear slope of 0.698532 (an indicator of an exogenous source) is significantly lower (99% confidence intervals of 0.69799\u0026ndash;0.699066) than that of seawater (~\u0026thinsp;0.7091) or basalt (0.7035), suggesting that neither sea spray nor basalt is an important exogenous source. Source inference based on \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios is likely incorrect.\u003c/p\u003e \u003cp\u003eThe 1/denominator form of a mixing diagram where 1/\u003csup\u003e86\u003c/sup\u003eSr is plotted against the observed and \u003cem\u003ey\u003c/em\u003e-intercept adjusted ratios is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ed. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ed also shows a slightly changing spline smoothed cubic derivative (small gray points) and the specific \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr derivative values (large gray points) associated with known \u003csup\u003e86\u003c/sup\u003eSr values at the five depths. This change in derivative (increasing with increasing 1Sr/\u003csup\u003e86\u003c/sup\u003eSr or decreasing with increasing \u003csup\u003e86\u003c/sup\u003eSr) is what one would expect from the subtle but not statistically significant difference in the \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr plots in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003eb. However, for an evaluation of the source, the relationship between \u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e87\u003c/sup\u003eSr is essentially linear.\u003c/p\u003e \u003cp\u003eThe slope of the mixing line (0.00000007105) in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ed approximates the \u003cem\u003ey\u003c/em\u003e-intercept (0.000000071786) of the original \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr plot shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003eb. The \u003cem\u003ey\u003c/em\u003e-intercept of this mixing line (0.69859) closely matches the slope (0.69853) shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003eb. Therefore, either a mixing diagram or an \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr plot can be used to identify exogenous sources and relative differences in backgrounds. A mixing diagram constructed from \u003cem\u003ey\u003c/em\u003e-intercept adjusted points is also shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ed. The \u003cem\u003ey\u003c/em\u003e-intercept adjustments were made as described above (Section 3.1 and Supplementary Material 1 in ref 1). This process eliminates \u003cem\u003ey\u003c/em\u003e-intercept effects and produces a flat mixing line if the original \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr plot is indeed linear. In this case, the flattened mixing line produces an almost identical source (\u003cem\u003ey\u003c/em\u003e-intercept) as the original mixing line. Individual points on this flattened mixing line represent the source estimates for individual depths. Additionally, Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ed shows the spline smoothed cubic derivative, which includes the \u003cem\u003ed\u003c/em\u003e\u003csup\u003e87\u003c/sup\u003eSr/\u003cem\u003ed\u003c/em\u003e\u003csup\u003e86\u003c/sup\u003eSr derivative values associated with known \u003csup\u003e86\u003c/sup\u003eSr values at five depths. This change in derivative (increasing with increasing 1/\u003csup\u003e86\u003c/sup\u003eSr or decreasing with increasing \u003csup\u003e86\u003c/sup\u003eSr) is what one would expect from the subtle but not statistically significant concave curvature in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003eb. All three source estimates in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ed (slope, modified mixing line with 1/\u003csup\u003e86\u003c/sup\u003eSr on the \u003cem\u003ey\u003c/em\u003e-axis, and a similar \u003cem\u003ey\u003c/em\u003e-intercept modified ratio mixing line) are in agreement. A traditional 1/total Sr mixing line also produces a similar source estimate (0.698594 data not shown).\u003c/p\u003e \u003cp\u003eAll five alternate source assessments shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ee (traditional 1/total Sr mixing line (\u003cem\u003ex\u003c/em\u003e-axis; \u0026micro;g/g units); 1/\u003csup\u003e86\u003c/sup\u003eSr form of a mixing line (\u003cem\u003ex\u003c/em\u003e-axis; \u0026micro;gmole/g units); \u003cem\u003ey\u003c/em\u003e-intercept corrected mixing line; \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr slope; and spline smoothed cubic derivative) plot against soil depth. Seawater is an unlikely source. The portion of Sr derived from the seawater calculations presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ea is based on \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios confounded by nonzero \u003cem\u003ey\u003c/em\u003e-intercepts. One might suspect that the derivative analysis in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ee, which detected the subtle but not statistically significant concave \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr relationship curvature, was the most accurate. The \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr relationship is not perfectly described by a linear function. The small \u003cem\u003ex\u003c/em\u003e-axis scale reveals apparent source pattern differences with soil depth that are likely unimportant.\u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ef, the five different source estimates (all shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ee) are plotted vs. depth along with the original \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr observed ratios, which require an expanded \u003cem\u003ex\u003c/em\u003e-axis scale. On the expanded scale, differences among the five source estimates are indeed unimportant. Seawater spray was clearly not the source of Sr in this example. An unknown exogenous material with a smaller \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio than the basalt parent material must be the source of these artificial data. Although not crucial in interpreting this artificial example, it is prudent to investigate whether apparently linear \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr relationships might have subtle curvatures. In the original unmodified dataset (Supplementary Material 1), possible curvatures in the \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr relationship are likely interpretively important.\u003c/p\u003e \u003cp\u003eIn some studies, changes in an observed isotopic ratio are likely due to changes in the isotopic ratio of material being added or removed from a system with different initial isotopic compositions. However, whether this is true cannot be determined in most isotopic research reports because they do not include total element data.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2. Untangling Confounded Data: A Sample Analysis of a More Complex Changing Source Real Data Example\u003c/h2\u003e \u003cp\u003eThe main objective of our first changing source example\u003csup\u003e15\u003c/sup\u003e was to investigate whether and how a speleothem-derived \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio and high-resolution total Sr record are correlated with the past climate and environment in the East Asian summer monsoon (EASM) region during the great climate shift associated with the last deglaciation. Samples were collected from a speleothem (stalagmite) in Songjiang Cave, Northeast Sichuan Province, Central China.\u003c/p\u003e \u003cp\u003eZhou \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e15\u003c/sup\u003e present cave data from central China for total Sr at various stalagmite depths that represent deposits between 20,000 and 10,000 years before present. We separated the data into three different time periods corresponding to three different depths (10\u0026ndash;12.25 shallow; 12.5\u0026ndash;14.25 deep; and 14.25\u0026ndash;20 thousand years deep before present), as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ea.\u003c/p\u003e \u003cp\u003eChanges in total Sr could drive differences in \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios. If \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr relationships have significant curvature and/or significant nonzero \u003cem\u003ey\u003c/em\u003e-intercepts, \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios can vary with total Sr in ways that are counterintuitive and difficult to predict\u003csup\u003e1\u003c/sup\u003e. The \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios for the same depths are presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003eb. These time divisions are based on stable \u003csup\u003e18\u003c/sup\u003eO:\u003csup\u003e16\u003c/sup\u003eO ratios that correspond to the ice core layer dates described below.\u003c/p\u003e \u003cp\u003eThe original authors include examples of \u003csup\u003e18\u003c/sup\u003eO data from several speleothems\u003csup\u003e16\u0026ndash;18\u003c/sup\u003e and compare these patterns to samples from Greenland ice cores. Temporal patterns differ. One would predict that different speleothem samples would differ in total O content since CaCO\u003csub\u003e3\u003c/sub\u003e concentrations and other sources of O differ for different speleothem layers at a single location and for similar times at different locations. Thus, speleothem \u003csup\u003e18\u003c/sup\u003eO:\u003csup\u003e16\u003c/sup\u003eO ratios that are confounded by total O differences will produce temporal patterns that differ.\u003c/p\u003e \u003cp\u003eHowever, ice core samples will have constant O concentrations for different layers or sites because the O content of water is always the same. There is no scaling of isotopic ratios if the denominators for different samples have constant total elemental concentrations. The ice core data copied from the Zhou \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e15\u003c/sup\u003e publication (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ec) reveal a decrease followed by an increase followed by stable \u003csup\u003e18\u003c/sup\u003eO:\u003csup\u003e16\u003c/sup\u003eO ratios that correspond to generally accepted changes in climate that are associated with the ice core layer dates.\u003c/p\u003e \u003cp\u003eThe original authors suggest simple binary mixing of two endmembers, the host rock of late Permian limestone, with a relatively lower \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio (\u0026sim;0.7071) and an exogenous Sr source with a relatively high \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio (\u0026sim;0.7109 deduced from 1/total Sr (\u003cem\u003ex\u003c/em\u003e-axis) vs. \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio (\u003cem\u003ey\u003c/em\u003e-axis) plots). We hypothesize that although mixing diagram equations similar to previously published results can be derived from a 1/\u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio plot, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ed (upper left), there are likely three different sources and backgrounds (also shown in the upper right in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ed) for the different depths that correspond to the time periods and climate differences revealed by changes in ice core oxygen isotopes (also shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ed).\u003c/p\u003e \u003cp\u003eThe nonlinear mixing lines (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ed) for the shallow and transition depths suggest that the sources for these two depths are not constant. The shallow pattern is as expected for concave \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr relationships with negative \u003cem\u003ey\u003c/em\u003e-intercepts\u003csup\u003e1\u003c/sup\u003e. The transition pattern is as expected for convex \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr relationships with positive \u003cem\u003ey\u003c/em\u003e-intercepts\u003csup\u003e1\u003c/sup\u003e. Indeed, the \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr relationships are statistically concave (negative \u003cem\u003ey\u003c/em\u003e-intercept), convex (positive \u003cem\u003ey\u003c/em\u003e-intercept) and linear (negative \u003cem\u003ey\u003c/em\u003e-intercept) for shallow (early), transitional, and late (deep) times, respectively (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ee). Although a combined linear function (upper right in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ed) also fits the data and approximates the authors\u0026rsquo; original conclusion, a constant exogenous source is unlikely.\u003c/p\u003e \u003cp\u003eThe deep data do not have a statistically significant curvature; thus, a linear function with a statistically significant negative \u003cem\u003ey\u003c/em\u003e-intercept describes the relationship shown. Because the two polynomial equations (early and transition) have either negative or positive \u003cem\u003ey\u003c/em\u003e-intercepts, the backgrounds (new starting points) also differ. Mixing diagrams (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ed), curvature analysis of the \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr plots (whether the x\u003csup\u003e2\u003c/sup\u003e term for a best-fit polynomial function is significant) (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ee) and an analysis of \u003cem\u003ed\u003c/em\u003e\u003csup\u003e87\u003c/sup\u003eSr/\u003cem\u003ed\u003c/em\u003e\u003csup\u003e86\u003c/sup\u003eSr derivatives (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ef) produce similar interpretations.\u003c/p\u003e \u003cp\u003eEarly time periods (shallow) suggest that \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios are becoming less enriched. Transition time periods are becoming more enriched, and late time periods (deep) have an unchanging exogenous source. The \u003csup\u003e18\u003c/sup\u003eO:\u003csup\u003e16\u003c/sup\u003eO ratio time patterns for the ice core (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ec) and stalagmite \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio time patterns (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003eb) are very different. However, the stalagmite derivative (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ef) and the unconfounded \u003csup\u003e18\u003c/sup\u003eO:\u003csup\u003e16\u003c/sup\u003eO ice core sample (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003ec) are in good agreement. Unfortunately, there are insufficient data (total elements not provided) to fully evaluate additional speleothem \u003csup\u003e18\u003c/sup\u003eO:\u003csup\u003e16\u003c/sup\u003eO data\u003csup\u003e16\u0026ndash;18\u003c/sup\u003e presented by the original authors. Researchers collected excellent data. With a little additional effort, the conclusions of their study are much more powerful. The results and interpretations from their stalagmite study closely match well-accepted interpretations of an ice core sample from Greenland.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3. Summary of Results\u003c/h2\u003e \u003cp\u003e \u003cb\u003eThe data\u003c/b\u003e from twelve well\u003cb\u003e-performed\u003c/b\u003e studies were selected as examples. A summary of these results is presented below.\u003c/p\u003e \u003cdiv id=\"Sec6\" class=\"Section3\"\u003e \u003ch2\u003e\u003cem\u003e2.3\u003c/em\u003e.1. Spelothem Sr isotopes and climate change\u003c/h2\u003e \u003cp\u003eResults from \u003cem\u003eZhou et al.\u003c/em\u003e\u003csup\u003e\u003cem\u003e15\u003c/em\u003e\u003c/sup\u003eCovered above Section \u003cspan refid=\"Sec4\" class=\"InternalRef\"\u003e2.2\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section3\"\u003e \u003ch2\u003e2.3.2. Sr isotopes and migration of ancient humans \u0026ndash; details are provided in Supplementary Material 1\u003c/h2\u003e \u003cp\u003eMontgomery and Evans\u003csup\u003e13\u003c/sup\u003e are correct. Resolving migration in human archaeological populations (Indigenous Machair dwellers and Immigrant Silica dwellers) with strontium mixing diagrams is required to define sources. Without a mixing diagram analysis, there would be little to report other than the mean \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios being similar for both populations, with Immigrant Silicate dwellers having more variability. Researchers would wrongly conclude that both populations were exposed to similar \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr sources.\u003c/p\u003e \u003cp\u003e \u003cem\u003e2.3.3. Details of the Sr isotopes and migration of an ancient cow are provided in Supplementary Material 1.\u003c/em\u003e \u003c/p\u003e \u003cp\u003eHorstwood \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e19\u003c/sup\u003e stated that the change in the direction of \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr with tooth age is opposite to what might be expected if the animal (a cow) moved from a high \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr source ratio to a lower one. Therefore, the authors suggest that the animal was likely to be slaughtered soon after moving to the low-\u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr source before any tooth changes could occur. We agree with the conclusion of slaughter soon after arrival but suspect that there was no change in diet as the original authors suggested. The variation in \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios from the cusp to the cervix that reflects early growth periods for the animal is entirely explained by differences in total Sr. There was likely no change in diet as the tooth developed.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section3\"\u003e \u003ch2\u003e2.3.4. Sr isotopes and migration of a prehistoric mammoth \u0026ndash; Details in Supplementary Material 1\u003c/h2\u003e \u003cp\u003eIsotopic researchers studying migration often assume that an observed isotopic ratio in a tissue reflects the source of a system, but this assumption is often not true. Sources derived from mammoth tooth data \u003cem\u003eKowalik et al.\u003c/em\u003e\u003csup\u003e\u003cem\u003e20\u003c/em\u003e\u003c/sup\u003e, rather than observed ratios, should have been used to make matches with geographic areas. Furthermore, changing source patterns in an animal will be difficult when matching an unchanging source value for a specific location. Source evaluations for the time period of interest must be constant. This was not the case in the data evaluated.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section3\"\u003e \u003ch2\u003e2.3.5. Tracking Sr isotope changes in a watershed \u0026ndash; Details in Supplementary Material 1\u003c/h2\u003e \u003cp\u003eAs theoretically demonstrated\u003csup\u003e1\u003c/sup\u003e, differences in the patterns of observed \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios and sources in a water inlet \u003cem\u003eBailey et al.\u003c/em\u003e\u003csup\u003e\u003cem\u003e21\u003c/em\u003e\u003c/sup\u003e can be explained by changes in total Sr. The time course variability is greater for sources than for observed ratios. This is expected since a large isotopic pool size buffers the observed ratio changes.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section3\"\u003e \u003ch2\u003e\u003cem\u003e2.3.6.\u003c/em\u003e Sea spray as a possible Sr source in soils. \u003cem\u003e\u0026ndash; Details in Supplementary Material 1\u003c/em\u003e\u003c/h2\u003e \u003cp\u003eWhipkey \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e14\u003c/sup\u003e are likely correct that seawater could be a Sr source in this Hawaiian soil example. However, conventional techniques overestimate the magnitude of sea spray input. Contrary to what the authors propose, the portion of Sr derived from seawater is both small and does not markedly decrease among the deeper depths.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section3\"\u003e \u003ch2\u003e\u003cem\u003e2.3.7.\u003c/em\u003e Nitrogen isotopes and trophic levels \u003cem\u003e\u0026ndash; details are provided in Supplementary Material 2\u003c/em\u003e\u003c/h2\u003e \u003cp\u003e \u003cem\u003eMoyo et al.\u003c/em\u003e \u003csup\u003e \u003cem\u003e22\u003c/em\u003e \u003c/sup\u003e suggested that sparrows occupy a slightly greater trophic level (~\u0026thinsp;0.7 delta \u003csup\u003e15\u003c/sup\u003eN unit difference) than marsh rats. However, when the five proposed alternate source methods were averaged, the sparrows and marsh rats were more different than the simple \u003csup\u003e15\u003c/sup\u003eN:\u003csup\u003e14\u003c/sup\u003eN ratios suggested. Sparrows have a \u003csup\u003e15\u003c/sup\u003eN:\u003csup\u003e14\u003c/sup\u003eN source ratio of 0.003759 vs. 0.003737 for marsh rats (an ~\u0026thinsp;2.7 delta unit difference). Conventional trophic level assignment also overestimates the trophic level assigned to phytoplankton.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section3\"\u003e \u003ch2\u003e\u003cem\u003e2.3.8.\u003c/em\u003e Nitrogen isotopes in a greenhouse tracer study \u003cem\u003e\u0026ndash; Details in Supplementary Material 2\u003c/em\u003e\u003c/h2\u003e \u003cp\u003eSandrock \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e23\u003c/sup\u003e suggested that the significant differences in the \u003csup\u003e15\u003c/sup\u003eN:\u003csup\u003e14\u003c/sup\u003eN ratio and NDFF for \u003cem\u003eE. alatus\u003c/em\u003e are caused by scaling effects rather than differences in physiology. These differences are indirectly caused by the smaller total N composition associated with a smaller plant size. One should not conclude that \u003cem\u003eE. alatus\u003c/em\u003e differs in its physiology and is fundamentally more efficient than the other two species evaluated.\u003c/p\u003e \u003cp\u003e \u003cem\u003e2.3.9.\u003c/em\u003e Evaluation of C isotopes on Mars as evidence for past life \u003cem\u003e\u0026ndash; Details in Supplementary Material 2\u003c/em\u003e\u003c/p\u003e \u003cp\u003eHouse \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e24\u003c/sup\u003e proposed three explanations for depleted Martian \u003csup\u003e13\u003c/sup\u003eC values: biological methane production, photoreduction of atmospheric CO\u003csub\u003e2\u003c/sub\u003e, and cosmic dust. All three of these explanations are unconventional and differ from processes common on Earth. Although all sediments have samples with depleted observed \u003csup\u003e13\u003c/sup\u003eC:\u003csup\u003e12\u003c/sup\u003eC ratios, it is striking that data from the oldest sediments produce evidence for depleted \u003csup\u003e13\u003c/sup\u003eC sources where the three younger formations do not. As is the case with climate change, migration, trophic level, and tracer studies discussed above, sources rather than observed ratios are what should be evaluated in searches for past life on Mars.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section3\"\u003e \u003ch2\u003e\u003cem\u003e2.3.10.\u003c/em\u003e Evaluation of C isotopes on Mars \u003cem\u003e\u0026ndash; Details in Supplementary Material 2\u003c/em\u003e\u003c/h2\u003e \u003cp\u003eHouse \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e24\u003c/sup\u003e also evaluated S isotopes for the same sediments on the basis of their evidence from past life studies conducted on the Martian samples described above. Total S confounds \u003csup\u003e34\u003c/sup\u003eS:\u003csup\u003e32\u003c/sup\u003eS evaluations of the same sediments.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section3\"\u003e \u003ch2\u003e\u003cem\u003e2.3.11.\u003c/em\u003e Challenging N isotope fractionation in ocean sediments \u003cem\u003e\u0026ndash; Details are provided in Supplementary Material 3.\u003c/em\u003e\u003c/h2\u003e \u003cp\u003eConstant negative Rayleigh N fractionation factors suggesting preferences for \u003csup\u003e14\u003c/sup\u003eN, the common isotope, which was calculated for sediments in the Aegean Sea\u003csup\u003e25,\u003c/sup\u003e are challenging. The significant ln total N vs delta \u003csup\u003e15\u003c/sup\u003eN slopes that define fractionation are driven by covarying total N. Issues are apparent for data with both linear and curvilinear \u003csup\u003e14\u003c/sup\u003eN vs. \u003csup\u003e15\u003c/sup\u003eN plots. Apparent fractionation can have alternate explanations. There may not be a constant microbial preference for lighter common isotopes defined by a fractionation factor. Rather, significant Rayleigh fractionation factors could be caused by relatively constant \u003csup\u003e15\u003c/sup\u003eN:\u003csup\u003e14\u003c/sup\u003eN loss ratios even though substrate isotopic ratios vary with depth.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section3\"\u003e \u003ch2\u003e\u003cem\u003e2.3.12.\u003c/em\u003e Challenging C isotope fractionation in soils \u003cem\u003e\u0026ndash; Details in Supplementary Material 3\u003c/em\u003e\u003c/h2\u003e \u003cp\u003eConstant negative Rayleigh C fractionation factors\u003csup\u003e\u003cem\u003e26\u003c/em\u003e\u003c/sup\u003e suggesting preferences for \u003csup\u003e12\u003c/sup\u003eC, the common isotope, which was calculated for soils on the Himalayan Plateau, are challenging. The significant ln total C vs delta \u003csup\u003e13\u003c/sup\u003eC slopes that define fractionation are driven by covarying total C. Issues are apparent for the linear \u003csup\u003e12\u003c/sup\u003eC vs. \u003csup\u003e13\u003c/sup\u003eC plots that were evaluated. Apparent fractionation can have alternate explanations. There may not be a constant microbial preference for lighter common isotopes defined by a fractionation factor. Rather, significant Rayleigh fractionation factors could be caused by relatively constant \u003csup\u003e13\u003c/sup\u003eC:\u003csup\u003e12\u003c/sup\u003eC loss ratios even though substrate isotopic ratios vary with depth.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"3. Discussion","content":"\u003cp\u003eIn summary, total Sr confounds \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios. Specifically, source evaluations rather than analyses of ratio-based expressions allow one to better 1) utilize \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios in migration patterns for ancient humans and animals, 2) associate speleothem \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr isotope data with known climate changes, 3) define the dynamics of \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr isotope data within a watershed and 4) define Sr sources in a soil. Three \u003csup\u003e15\u003c/sup\u003eN:\u003csup\u003e14\u003c/sup\u003eN, two \u003csup\u003e13\u003c/sup\u003eC:\u003csup\u003e12\u003c/sup\u003eC and one \u003csup\u003e34\u003c/sup\u003eS:\u003csup\u003e32\u003c/sup\u003eS studies were also re-evaluated. Total N confounds the interpretation of \u003csup\u003e15\u003c/sup\u003eN:\u003csup\u003e14\u003c/sup\u003eN studies on 1) isotopic fractionation in the Aegean Sea, 2) N tracers in potted plants and 3) trophic level studies in a marsh. Total C confounds \u003csup\u003e13\u003c/sup\u003eC:\u003csup\u003e12\u003c/sup\u003eC studies on 1) isotopic fractionation in soil and 2) an evaluation of whether past life existed in Martian sediments collected by the Curiosity rover. Total S confounds \u003csup\u003e34\u003c/sup\u003eS:\u003csup\u003e32\u003c/sup\u003eS studies in the same Martian sediments.\u003c/p\u003e \u003cp\u003eIn much of the ratio literature, nonzero \u003cem\u003ey\u003c/em\u003e-intercepts in denominator vs. numerator plots mathematically explain why a ratio is related to denominator size\u003csup\u003e8,29\u003c/sup\u003e. However, nonzero \u003cem\u003ey\u003c/em\u003e-intercepts are counterintuitive and conceptionally awkward. For example, a positive \u003cem\u003ey\u003c/em\u003e-intercept for an \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr relationship could suggest that as total Sr (or the closely related \u003csup\u003e86\u003c/sup\u003eSr) disappears, some \u003csup\u003e87\u003c/sup\u003eSr still exists. Why would some \u003csup\u003e87\u003c/sup\u003eSr still be present when total Sr is zero? Similarly, a negative \u003cem\u003ey\u003c/em\u003e-intercept implies that a negative amount of \u003csup\u003e87\u003c/sup\u003eSr exists as the total Sr approaches zero. How can there be a negative value for an isotope? This inconsistency is explained by the fact that nonzero \u003cem\u003ey\u003c/em\u003e-intercepts have chemical meaning. For both linear and polynomial denominators vs. numerators, isotopic functions \u003cem\u003ey\u003c/em\u003e-intercepts are related to backgrounds.\u003c/p\u003e \u003cp\u003eBecause they are not normally distributed, Isles\u003csup\u003e6\u003c/sup\u003e suggested that using raw (nontransformed) ratios to interpret elemental ratio data is inappropriate. He suggested that log-transformed or geometric means should be used instead and provides an example where nontransformed elemental N:P ratios provide misleading interpretations for very large datasets with a broad range of N:P ratios. Although we agree with Isles\u0026rsquo;s\u003csup\u003e6\u003c/sup\u003e concerns that ratio-based expressions are often misleading, these specific issues do not apply to isotopic ratios that vary over a narrow range. However, before evaluating the raw isotopic ratios and the slopes and \u003cem\u003ey\u003c/em\u003e-intercepts of plots of their denominators and numerators, we first determined that using arithmetic means, log-transformed means, or geometric means of isotopic ratios led to the same interpretation of differences among times, depths, or treatments. We have not found any cases where Isles\u0026rsquo;s\u003csup\u003e6\u003c/sup\u003e transformation concerns apply to isotopic ratios (79Sr datasets, including examples here and in Supplementary Material S; data not shown but available on request).\u003c/p\u003e \u003cp\u003eArguments have also been presented where the use of log-transformed data has been challenged (the opposite of Isles\u0026rsquo;s\u003csup\u003e6\u003c/sup\u003e concerns expressed above). The great influence of scaling studies has attracted scrutiny as to the validity of using log-transformed data when evaluating scaling relationships\u003csup\u003e30\u0026ndash;33\u003c/sup\u003e. Each of these publications has been challenged\u003csup\u003e34\u0026ndash;37\u003c/sup\u003e. We agree with these challenges when the plotted values range over several orders of magnitude. However, there are still many advantages to using nontransformed data. We therefore advocate the use of nontransformed data to evaluate isotopic scaling.\u003c/p\u003e \u003cp\u003eNontransformed data are especially useful because determining whether denominator vs. numerator plots are linear is important. One wants to know if sources are constant or changing. A consistent curvature in a raw data plot can be verified with significant \u003cem\u003ex\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e terms in the polynomial denominator vs. numerator analyses or nonlinear mixing diagrams. All the artificial perfectly linear functions in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e1\u003c/span\u003ea and c in our companion paper\u003csup\u003e1\u003c/sup\u003e that have nonzero \u003cem\u003ey\u003c/em\u003e-intercepts can be approximated with log-log functions that have r\u003csup\u003e2\u003c/sup\u003e values greater than 0.999999 with slopes that significantly differ from 1.0 (data not shown). However, in all the cases, the implied curvatures indicated by the significant power functions fit truly linear functions, suggesting that changing sources do not exist. Slopes for nontransformed linear denominator vs. numerator plots define sources. The nonzero \u003cem\u003ey\u003c/em\u003e-intercepts for linear or polynomial nontransformed denominator vs. numerator functions provide interpretive information regarding backgrounds. This being said, we see no harm in using either approach (as presented in this report) to glean more information.\u003c/p\u003e \u003cp\u003eOne should always evaluate traditional Keeling plots, other forms of mixing diagrams, or an analysis of the relationship between the ratio denominator and the numerator. Source evaluations inform researchers when isotopic ratios are confounded. Many datasets may not have ratio issues; thus, ratio-based assessments can still be useful. However, when total element data are available, finding confounded ratio examples is not difficult. One will never know if isotopic data are confounded unless they are evaluated.\u003c/p\u003e"},{"header":"4. Methods","content":"\u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003e4.1. Untangling Confounded Data: Five Approaches\u003c/h2\u003e \u003cp\u003eIsotopic ratios that are confounded by different denominator sizes will be misleading. There are many ways to compensate for the errors dictated by mathematical theory. The five procedures proposed here allow one to define the relationship between numerators and denominators more clearly:\u003c/p\u003e \u003cp\u003e1) \u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003ePlotting the denominator vs. the numerator concentrations (both in umole/g units) will produce a slope that is equal to the source (m in the\u003c/span\u003e \u003cspan type=\"ItalicUnderline\" class=\"ItalicUnderline\" name=\"Emphasis\"\u003ey\u003c/span\u003e\u0026thinsp;\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e=\u0026thinsp;mx\u0026thinsp;+\u0026thinsp;b equation for a denominator vs. numerator plot) for linear functions.\u003c/span\u003e The denominator vs. numerator plot can be evaluated to determine whether a source is consistently increasing or decreasing by evaluating the significance of the \u003cem\u003ex\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e term of best-fit polynomial equations.\u003c/p\u003e \u003cp\u003e2) \u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eAdjusting standard isotopic ratios (always calculated on a molar basis) to account for nonzero\u003c/span\u003e \u003cspan type=\"ItalicUnderline\" class=\"ItalicUnderline\" name=\"Emphasis\"\u003ey\u003c/span\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e-intercepts removes the background and total element effects that confound interpretation of sources.\u003c/span\u003e The value of a positive \u003cem\u003ey\u003c/em\u003e-intercept for a denominator vs. numerator plot (in umolar units) can be subtracted from the numerator isotope values of a dataset before \u0026lsquo;new\u0026lsquo; ratios are recalculated with the original denominators. Similarly, the absolute value of a negative \u003cem\u003ey\u003c/em\u003e-intercept can be added to the numerator isotope values of a dataset, after which a newly calculated ratio can be calculated from the original data. For polynomial denominator vs. numerator functions, a second \u003cem\u003ey\u003c/em\u003e-intercept correction method uses a feature of polynomial equations where the ratios for a function with a zero intercept can be calculated from the terms of the original \u003csup\u003e86\u003c/sup\u003eS vs. \u003csup\u003e87\u003c/sup\u003eSr relationships, regardless of whether the original function has a zero \u003cem\u003ey\u003c/em\u003e-intercept\u003csup\u003e1\u003c/sup\u003e. For both linear and curvilinear functions, procedures should not be viewed as an extrapolation of a value beyond the range of points presented by the data. Instead, a \u003cem\u003ey\u003c/em\u003e-intercept correction for a linear function can be viewed as indirectly and mathematically collecting points that define the constant derivative (source) for the range of points evaluated. This defines a source for each individual data point. An average of all \u003cem\u003ey\u003c/em\u003e-intercept adjusted points approximates the slope of the original linear denominator vs. the numerator function. For nonlinear functions, adjusted \u003cem\u003ey\u003c/em\u003e-intercepts do not produce a collection of points that define the derivative; rather, these points define the pattern of ratio changes as the denominator increases for a curvilinear function with a zero \u003cem\u003ey\u003c/em\u003e-intercept.\u003c/p\u003e \u003cp\u003e3) \u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eThe\u003c/span\u003e \u003cspan type=\"ItalicUnderline\" class=\"ItalicUnderline\" name=\"Emphasis\"\u003ey\u003c/span\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e-intercept defines the source (molar-based isotopic ratio) for traditional mixing diagram plots where 1/total element concentration (usually expressed as \u0026micro;g/g or ppm) is plotted vs. isotopic ratio (calculated on a molar basis)\u003c/span\u003e. This occurs because, at an infinite elemental concentration (1/element concentration approaches zero), the effect of the original background has been diluted, and the observed isotopic ratio approaches the source of material being added.\u003c/p\u003e \u003cp\u003e4) \u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eThe\u003c/span\u003e \u003cspan type=\"ItalicUnderline\" class=\"ItalicUnderline\" name=\"Emphasis\"\u003ey\u003c/span\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e-intercept defines the source (molar-based isotopic ratio) as follows: 1/denominator (umole/g units) is used as the\u003c/span\u003e \u003cspan type=\"ItalicUnderline\" class=\"ItalicUnderline\" name=\"Emphasis\"\u003ex\u003c/span\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e-axis rather than the traditional 1/total element concentration.\u003c/span\u003e The modified mixing diagram is mathematically linked to the denominator vs. numerator function. If the \u003cem\u003ex\u003c/em\u003e-axis is expressed in 1/umole/g units, the slope of a modified (1/denominator vs. isotopic ratio) mixing diagram will equal the original denominator vs. numerator \u003cem\u003ey\u003c/em\u003e-intercept, and the mixing diagram \u003cem\u003ey\u003c/em\u003e-intercept will equal the original denominator vs. numerator slope. With less than perfect real data, the relationships between the modified mixing diagram slope and the original denominator and the numerator \u003cem\u003ey\u003c/em\u003e-intercept approximate one another but may not be exact matches.\u003c/p\u003e \u003cp\u003e5) \u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eDerivatives can be calculated from the denominator vs. numerator data.\u003c/span\u003e For a given denominator, the derivative represents the source (molar isotopic ratio) if both the numerator and denominator are calculated on a molar basis. A flat derivative vs. denominator concentration plot suggests that the sources are constant. A changing derivative suggests changing sources. There are many ways to calculate a numeric derivative. Here, we used a simple spine smoothing cubic option in SYSTAT TableCurve\u003csup\u003e\u0026reg;\u003c/sup\u003e 2D software. The software can produce derivative values for a wide range of isotopic values in the original dataset. This information can be used to produce derivative values (interpolation for an estimation of source) for every observation.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec19\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Overview of Data Examples\u003c/h2\u003e \u003cp\u003eWe have supplemented past theoretical explanations, artificial data examples, simulations and companion \u003csup\u003epaper 1\u003c/sup\u003e with detailed analyses of Sr examples from both a modified dataset (Section \u003cspan refid=\"Sec3\" class=\"InternalRef\"\u003e2.1\u003c/span\u003e) and the cave data (Section \u003cspan refid=\"Sec4\" class=\"InternalRef\"\u003e2.2\u003c/span\u003e) discussed above. Additional \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr data from the following publications are presented: (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e) Montgomery and Evans\u003csup\u003e13\u003c/sup\u003e, (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e) Horstwood \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e19\u003c/sup\u003e, (\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e) Kowalik \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e20,\u003c/sup\u003e (\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e) Zhou \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e15\u003c/sup\u003e and (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e) Bailey \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e21\u003c/sup\u003e Publications (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e)-(\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e) refer to studies on Sr sources in human teeth, migration possibilities for an ancient cow, migration possibilities for a European mammoth, a speleothem, and water from a river inlet, respectively. Another dataset, from Whipkey \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e14,\u003c/sup\u003e that addresses Sr sources in soil is presented in Supplementary Material S2. These are the original data that were modified to create the first artificial example (Section \u003cspan refid=\"Sec3\" class=\"InternalRef\"\u003e2.1\u003c/span\u003e, Untangling confounded sources of a constant source artificial example).\u003c/p\u003e \u003cp\u003eData re-evaluations involving isotopic analyses of N, C and S from the following publications were also conducted: (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e) Moyo \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e22\u003c/sup\u003e, (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e) Sandrock \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e23\u003c/sup\u003e, and (\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e) House \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e24\u003c/sup\u003e Publication (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e) addresses C and N isotope evaluations to evaluate trophic levels (only nitrogen isotopes were used in our re-evaluation); Publication (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e) covers a \u003csup\u003e15\u003c/sup\u003eN-depleted tracer study on ornamental plants grown in pots; and Publication (\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e) involves an isotopic analysis of \u003csup\u003e13\u003c/sup\u003eC and \u003csup\u003e34\u003c/sup\u003eS on samples collected by the Curiosity rover on Mars.\u003c/p\u003e \u003cp\u003eData re-evaluations involving N\u003csup\u003e25\u003c/sup\u003e and C\u003csup\u003e26\u003c/sup\u003e isotopes are presented. M\u0026ouml;bius \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e25\u003c/sup\u003e evaluated N isotopes in Aegean Sea sediments. Wang \u003cem\u003eet al\u003c/em\u003e.\u003csup\u003e26\u003c/sup\u003e conducted a soil C isotope study on the Tibetan Plateau. Both re-evaluations challenge the existing concepts of isotopic fractionation. As stated above, the details of all the dataset re-evaluations are presented in Supplementary Materials S1-S3.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec20\" class=\"Section2\"\u003e \u003ch2\u003e\u003cem\u003e4.3 Methods for Initial Artificial (Modified Data) Linear Denominator\u003c/em\u003e vs. \u003cem\u003eNumerator Example\u003c/em\u003e\u003c/h2\u003e \u003cp\u003eMost \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios in the literature are recorded up to the 5th decimal, and percentages of total \u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e87\u003c/sup\u003eSr are reported to the 4th decimal in accordance with the National Institute of Standards and Technology Certificate of Analyses for the SRM 987 standard. All of the authors\u0026rsquo; work with Sr isotopes was re-evaluated here, and in Supplementary Material 2, we used this standard. The standard values are 0.71039\u0026thinsp;\u0026plusmn;\u0026thinsp;0.00013, 7.0015\u0026thinsp;\u0026plusmn;\u0026thinsp;0.0026, and 9.8566\u0026thinsp;\u0026plusmn;\u0026thinsp;0.0034 for the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio, percent \u003csup\u003e87\u003c/sup\u003eSr and percent \u003csup\u003e86\u003c/sup\u003eSr, respectively. Standard errors for the SRM 987 sample listed above are relatively high, but since mass spectrometers measure relative differences far more accurately than absolute differences reporting to the 5th decimal place when a reference sample is used is appropriate.\u003c/p\u003e \u003cp\u003eProducing \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr plots for real datasets requires some assumptions because only the total isotope concentration and \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios are known. Thus, not all isotopic species are accounted for. For the modified data, the portion for the sum of \u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e87\u003c/sup\u003eSr was forced to remain constant and was equal to the standard values of 9.8566% + 7.0015%. This assumption was more fully evaluated for the real datasets described below.\u003c/p\u003e \u003cp\u003eWhen total Sr concentrations are presented in the literature, they are usually expressed in ppm or \u0026micro;g/g units. These total concentration units can be converted to \u0026micro;m/ml or \u0026micro;m/g by assuming that the molecular weight, including all the Sr mass isotopes, is 87.62. The combined total for both \u003csup\u003e87\u003c/sup\u003eSr and \u003csup\u003e86\u003c/sup\u003eSr (in \u0026micro;mol/g units) can then be calculated (total * 0.168581). The values for both \u003csup\u003e87\u003c/sup\u003eSr and \u003csup\u003e86\u003c/sup\u003eSr (in \u0026micro;mol/g units) can then be calculated (total * 0.1685811). The \u003csup\u003e86\u003c/sup\u003eSr value\u0026thinsp;=\u0026thinsp;combined \u003csup\u003e87\u003c/sup\u003eSr and \u003csup\u003e86\u003c/sup\u003eSr (in \u0026micro;mol/g units/\u003csup\u003e87\u003c/sup\u003eSR:\u003csup\u003e86\u003c/sup\u003eSr ratio\u0026thinsp;+\u0026thinsp;1). The \u003csup\u003e87\u003c/sup\u003eSr value\u0026thinsp;=\u0026thinsp;the combined total of \u003csup\u003e87\u003c/sup\u003eSr and \u003csup\u003e86\u003c/sup\u003eSr \u0026ndash; 86Sr. In all the cases, the sums of the 87Sr\u0026thinsp;+\u0026thinsp;86Sr values were checked to ensure that the value was 0.168581 * total Sr in \u0026micro;m. The \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios (recalculated from the derived umolar/g values of \u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e87\u003c/sup\u003eSr) matched the original published \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios.\u003c/p\u003e \u003cp\u003eOnce the \u003csup\u003e87\u003c/sup\u003eSr and \u003csup\u003e86\u003c/sup\u003eSr concentrations are known, one can determine the statistical significance of the \u003cem\u003ex\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e terms in the polynomial best-fit \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr equations and whether the descriptive denominator vs. numerator functions are linear, convex (\u003csup\u003e87\u003c/sup\u003eSr increases with increasing \u003csup\u003e86\u003c/sup\u003eSr), or concave (\u003csup\u003e87\u003c/sup\u003eSr decreases with increasing \u003csup\u003e86\u003c/sup\u003eSr). Spline-smoothed derivatives can also be determined (with TableCurve\u0026reg; 2D Sigma software). It has been our experience that spline smoothed cubic derivatives are not always helpful because some datasets have considerable variation. However, in other cases, these derivatives are extremely useful in interpreting datasets and often closely agree with mathematical derivatives derived from a polynomial best-fit denominator vs. numerator equation. Knowing whether mixing lines are linear also provides insight into whether the original denominator vs. numerator plot has subtle curvature. Mathematically, a linear denominator vs. numerator function must produce a linear mixing diagram.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec21\" class=\"Section2\"\u003e \u003ch2\u003e4.4 Methods for All Real Data Examples\u003c/h2\u003e \u003cp\u003eReal \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr datasets were analyzed as described above. Changes in portions of specific isotopes in natural systems are extremely small, but it is important to determine whether slight differences in assumptions will alter the results and which assumptions are appropriate. The assumption that the sum of \u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e87\u003c/sup\u003eSr is equal to the standard value of 9.8566% + 7.0015% was validated by comparing the results of two other assumptions: 1) the portion of \u003csup\u003e86\u003c/sup\u003eSr remains constant and is equal to the standard value of 9.8566%, and 2) the portion of \u003csup\u003e87\u003c/sup\u003eSr remains constant and is equal to the standard value of 7.0015%.\u003c/p\u003e \u003cp\u003eThe 9.8566% + 1.0015% assumption produces intermediate \u003csup\u003e86\u003c/sup\u003eSR vs. \u003csup\u003e87\u003c/sup\u003eSr slope and \u003cem\u003ey\u003c/em\u003e-intercept values that are only slightly different than the results for assumptions 1) and 2) above. Therefore, the constant 87Sr n\u0026thinsp;+\u0026thinsp;86Sr assumption was used thereafter. The evaluation of all three assumptions in Supplemental Material 1, which addresses the migration of ancient humans\u003csup\u003e13,\u003c/sup\u003e suggests that the \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr slope, calculated with a constant \u003csup\u003e87\u003c/sup\u003eSr plus \u003csup\u003e86\u003c/sup\u003eSr assumption, produces an appropriate estimate of sources.\u003c/p\u003e \u003cp\u003eThe theoretical, simulated, and real data for Sr isotope evaluations (with the previously stated assumption) all suggest problems interpreting Sr isotope ratios. Without this Sr assumption, one will be forced to conclude that although cases where Sr isotope ratios are confounded by differences in total elements are rampant, there is no way to fully re-evaluate the Sr literature, calculate \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr slopes and derivatives, and determine whether sources are constant or statistically changing. However, although rarely used, traditional mixing diagram approaches using raw data can still be applied to define isotopic Sr sources if linear mixing lines are produced.\u003c/p\u003e \u003cp\u003eIn other analyses (Supplementary Materials 2 and 3), non-Sr isotope re-evaluation examples are presented. The data reanalyses were performed as explained in the initial Sr isotope examples explained above. However, no assumptions (required when an element has more than two isotopic species) were used when calculating denominator or numerator values for N and C studies. Nitrogen and C have only two major mass isotopes. Sulfur has more than two other isotope species. However, denominator vs. numerator plots were not constructed for S isotopes because traditional mixing diagrams easily define sources.\u003c/p\u003e \u003cp\u003eSlopes and derivatives reflect constant and changing sources, respectively. Changes in sources for different layers or samples over time can be determined for re-evaluated datasets by matching derivatives calculated from \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr relationships to specific values of \u003csup\u003e86\u003c/sup\u003eSr or total Sr. All new information was interpreted to determine if the original authors\u0026rsquo; conclusions were supported.\u003c/p\u003e \u003cp\u003eImportant details for all the isotopic data reanalyses are presented in the supplementary material (Supplemental Material 1, Data re-evaluation for SR isotopes; Supplemental Material 2, Data re-evaluations for N, C and S isotopes; and Supplemental Material 3, Rethinking fractionation favoring the common isotope.\u003c/p\u003e \u003c/div\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003ch2\u003eCompeting Interests\u003c/h2\u003e \u003cp\u003eThe author(s) declare no competing interests.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eTLR is the team leader and was responsible for developing concepts over many years. CPN and CN spent considerable time investigating ratio-related issues that included isotope evaluations during their Master of Science work under TLR that in part led to this publication. CPN and CN also made substantial editing contributions. KM was the major editing resource for the team and responsible for improving the initial rough draft of the manuscript produced by TLR. FC and DL made editing contributions that were focused on isotopic scaling issues directed at readers that had limited exposure to both isotopic research and ratio-related scaling issues.\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eSincere thanks and love of TLR\u0026rsquo;s life Mary Ann Righetti (BS,MLS,JD). She has been a sounding board for the many years of concept development. Also appreciated are a group of undergraduate students that edited this manuscript. We reasoned that for a publication this controversial to be accepted, every sentence in the text and figure legends should be understandable by a well-trained undergraduate. Students spent countless hours improving the manuscript. Their names are listed in supplemental material.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eAll data generated or analysed during this study are included in this published article [and its supplementary information files].\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eMoots, K. Nguyen, C. P Nguyen C. Camacho, F., Lindstrom, D., and Righetti, T, L.. A mathematical explanation for why ratio-based isotopic analyses are commonly misleading: Theory. \u003cem\u003eScientific Reports this issue\u003c/em\u003e (2024).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eAtchley, W. R., Gaskins, C. T. \u0026amp; Anderson, D. Statistical properties of ratios. I. Empirical results. 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Soc. 125, 664\u0026ndash;671 (2018).\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-4086527/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4086527/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eDozens of preliminary data reevaluations were conducted to verify the ratio-related mathematical theory. Differences in total elements among treatments, times and/or conditions frequently confound interpretation because total element values affect isotopic ratios. Eventually, twelve (six \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr, three \u003csup\u003e15\u003c/sup\u003eN:\u003csup\u003e14\u003c/sup\u003eN, two \u003csup\u003e13\u003c/sup\u003eC:\u003csup\u003e12\u003c/sup\u003eC and one \u003csup\u003e34\u003c/sup\u003eS:\u003csup\u003e32\u003c/sup\u003eS) well-performed studies were selected as examples. \u003cem\u003eSr studies:\u003c/em\u003e Source evaluations better describe migration patterns for ancient humans and animals, better align speleothem isotopic data with known climate changes, better define the dynamics of isotopic data within a watershed, and better describe sources of soil Sr. \u003cem\u003eN studies:\u003c/em\u003e Source evaluations change interpretations for isotopic fractionation in sediments; N tracer treatments on potted plants; and trophic level assignments for different species in a marsh. \u003cem\u003eC studies:\u003c/em\u003e Total C confounds \u003csup\u003e13\u003c/sup\u003eC:\u003csup\u003e12\u003c/sup\u003eC data for isotopic fractionation experiments in forest soils and complicates an evaluation of whether past life existed in Martian sediments collected by the Curiosity rover. \u003cem\u003eS studies\u003c/em\u003e: Total S also confounds \u003csup\u003e34\u003c/sup\u003eS:\u003csup\u003e32\u003c/sup\u003eS evaluations of the same Martian sediments. We intend to emphasize that source analyses provide better isotopic interpretations than observed ratios in agricultural, biological and environmental studies. Observed isotopic ratio changes do not necessarily reflect source changes. Source analyses improved the Sr, N, C and S isotope evaluations.\u003c/p\u003e","manuscriptTitle":"A Mathematical Explanation for Why Ratio-Based Isotopic Analyses are Commonly Misleading: Dealing with Confounded Isotopic Ratios","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-05-02 19:43:56","doi":"10.21203/rs.3.rs-4086527/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"1a0de814-dd2b-43de-bf7b-08838d93cd4f","owner":[],"postedDate":"May 2nd, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":31331497,"name":"Biological sciences/Biological techniques"},{"id":31331498,"name":"Earth and environmental sciences/Biogeochemistry"},{"id":31331499,"name":"Earth and environmental sciences/Climate sciences"},{"id":31331500,"name":"Earth and environmental sciences/Ecology"},{"id":31331501,"name":"Earth and environmental sciences/Environmental sciences"},{"id":31331502,"name":"Physical sciences/Chemistry"}],"tags":[],"updatedAt":"2024-05-22T04:31:29+00:00","versionOfRecord":[],"versionCreatedAt":"2024-05-02 19:43:56","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4086527","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4086527","identity":"rs-4086527","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}