A Mathematical Explanation for Why Ratio-Based Isotopic Analyses are Commonly Misleading: Theory

A Mathematical Explanation for Why Ratio-Based Isotopic Analyses are Commonly Misleading: Theory | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article A Mathematical Explanation for Why Ratio-Based Isotopic Analyses are Commonly Misleading: Theory Kate Moots, Christina P. Nguyen, Catherine Nguyen, Frank Camacho, and 2 more This is a preprint; it has not been peer reviewed by a journal.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-4086468/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 Stable mass isotopic ratios (such as 13 C: 12 C, 15 N: 14 N, 18 O: 16 O 87 Sr: 86 Sr and 34 S: 32 S) are used to interpret archaeological, climate change, ecological, geological, and physiological studies. Most isotopic reports evaluate changes in observed isotopic ratios or ratio-based expressions over time or among treatments. To address concerns that ratios or ratio-based expressions may not produce conclusions that support known physiological or ecological principles, source (isotopic ratio of the material being added or lost) analyses are proposed as an alternative to statistically analysing observed isotopic ratios. Mathematically defined relationships between observed ratios, backgrounds (isotopic ratio of a system before any loses or gains), sources and total element concentrations as well as denominator vs. numerator relationships are presented. These mathematical relationships suggest that source-based approaches often produce conclusions that differ from ratio-based evaluations. Total element concentrations, necessary for source analyses, are presented in less than half of isotopic publications. Without evaluating total element, relative background and source ratios cannot be determined. Even, when total element data is available, researchers rarely conduct source analyses. This is unfortunate because determining sources solves most interpretive issues. Our goal is to advocate better methods when analyzing isotopic ratios in the thousands of mass isotope publications annually produced. 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 Figure 3 1. Introduction Observed stable mass isotopes serve as a proxy for past climates 1 and are used as tracers in archaeological 2,3 , ecological 4 , geological 1 and plant or animal physiological studies ,5 (see Supplemental Material: Section 1 , Review of isotopic principles). These applications involve evaluations of observed isotopic ratios or ratio-based expressions (see Supplemental Material: Section 2 , Formulas for converting isotopic ratios to ratio-based expressions). Values of variables calculated from observed ratios are also similarly evaluated (see Supplemental Material: Section 3 , Formulas for converting isotopic ratios to portion of total element derived from an added source) for an example. This mathematical and statistical report emphasizes source (isotopic ratio of material being added or lost) analyses and challenges current methods for evaluating isotopes. Isotopic studies usually involve standard statistical evaluations of observed isotopic ratios or ratio-based expressions for different times, treatments, or conditions. These standard procedures have flourished despite warnings about using ratios in research studies being repeatedly reported for over a century 6 . Concerns that ratios or ratio-based isotopic expressions produce conclusions that do not support known physiological or ecological principles have been reported 7–15 . Twelve additional examples of isotopic evidence that contradict other known facts are presented in this two-paper series. The project leader (TLR) has also made interpretive mistakes. A re-evaluation of the first dataset that caught his attention is discussed 16 (see Supplementary Material 3 in reference 16). A review of the ratio literature on which this challenge is based is presented (see Supplemental Material: Section 4 , Review of ratio literature and isotopic warnings). We emphasize that observed isotopic ratios for a system will depend on sources, backgrounds (isotopic ratio of a system before any loses or gains) and total element concentration or content differences among different times, treatments, or conditions. Concepts demonstrated here are important because ratios are often misleading as they depend on denominator size 6,17–20 . Without measuring and analysing total element one cannot determine if changing observed isotopic ratios are caused by source changes or differing amounts of total element. We propose that source is the most important variable to analyse but, it is often ignored as statistical evaluations are mostly conducted on observed ratios. Our presentation may qualify as an extraordinary claim. How could research methodologies associated with scientific breakthroughs in many scientific disciplines have mathematical flaws? We are pursuing an “extraordinary claims require extraordinary evidence” philosophy championed by Sagan 21 . More mathematical descriptions, artificial examples, simulations and data re-evaluations are presented in the text and supplemental material for this two-paper series than what is usually presented. Studies as diverse as 87 Sr: 86 Sr evaluations assessing climate change from cave speleothems 16 ; 15 N: 14 N trophic level studies and 13 C: 12 C investigations focused on the possibility of previous life on Mars 16 (see Supplemental Material 2 in reference 16) suggest that source (loss or gain) evaluations and traditional approaches evaluating either isotopic ratios or ratio-based expressions produce very different interpretations. Current concepts related to isotopic fractionation are also challenged 16 (see Supplemental Materials 3 in reference 16). Our goal is to advocate better methods when analyzing isotopic ratios in the thousands of mass isotope publications annually produced. Evaluating sources, which requires knowledge of total element concentration, provides more detail than solely evaluating observed isotopic ratios. Keeling 22 plots are commonly used to determine sources in gaseous 13 C studies 23 . Variations of Keeling’s original procedure referred to as mixing diagrams for non-gas samples are also used 24 . Details of these procedures are presented in supplementary material (see Supplemental Material: Section 5, Quantifying isotopic sources and Supplemental Material: Section 6, More nuances of Keeling plots and mixing diagrams). A summary of 300 evaluations (100 each for 13 C, 15 N and 87 Sr studies) shown in Table 1 (see Supplemental Material: Section 7 for methods) demonstrates the magnitude of the problem. In our limited sample of gaseous carbon studies (6 of 100 13 C publications), mixing diagrams are used in half of the combined CO 2 and CH 4 isotopic research publications. Otherwise, sixty years after being introduced for gaseous studies, the use of Keeling’s approach is rare. Only 17% of Sr isotopic publications use source analyses. Although common for CO 2 and CH 4 research, less than 10% of all isotopic C studies use the technique. For N isotopic research publications none used source analyses. The total element concentrations, necessary for source analyses, are likely available in less than half of isotopic publications. Table 1 Portion of 15 N, 13 C, and 87 Sr publications that include information related to total element concentrations that can be matched with isotopic ratio data. Also included are the portion of total publications where a source analysis (Keeling plots or mixing diagrams) was utilized. Total element data not included % Total element data used to calculate sources % of all publications % of publications with total element Non-gas publications 15 N 57 0 0 13 C 79 3 14 87 Sr 45 17 30 Gas publications 15 N 61 0 0 13 C 33 50 75 All publications 15 N 65 0 0 13 C 70 9 30 87 Sr 45 17 30 2. Results 2.1 Mathematical Relationships for Ratio-Based Expressions in Artificially Created Linear Denominator vs. Numerator Relationships The first artificial datasets (Fig. 1 a-e) demonstrate the effect of very small differences in y -intercepts for linear denominator vs. numerator functions on observed 87 Sr: 86 Sr ratios. Plots of the umole/ml denominator vs. the umole/ml numerator of isotopic ratios as presented in Fig. 1 a and c are not evaluated in the isotopic literature. They are presented here to demonstrate the mathematical relationships between linear denominator vs. numerator plots and plots of the denominator or total isotope vs. an isotopic ratio. The datasets presented in Fig. 1 a-e, represent a system where Sr-containing material with a constant 87 Sr: 86 Sr ratio is being added to or removed from an existing system, where the slope value for an 86 Sr vs. 87 Sr plot is equal to the 87 Sr: 86 Sr ratio of modern seawater. For simplicity, this artificial data (and similar data in Fig. 1 c and d) is first imagined as coming from a system where Sr-containing material with a constant 87 Sr: 86 Sr ratio is being added to an existing system. Slopes of denominator vs. numerator regression equations will therefore be related to sources (the isotopic ratio of an element gained). The same system viewed as losing total element will be subsequently discussed. With the linear equations as a starting point, it is easier to take the next step and demonstrate that isotopic ratios must depend on their denominators and total isotope concentrations (Fig. 1 b and d). The slope of the 86 Sr vs. 87 Sr regression line is the isotopic ratio of the material being added to the system. This dataset results in a linear relationship when 86 Sr vs. 87 Sr are plotted because the ratio of 87 Sr: 86 Sr in the source remains constant. In Fig. 1 a, the existing system background has a higher 87 Sr: 86 Sr ratio than the material being added. Small deviations from an isometric relationship result in 3 infinitesimally different regression lines, each with very small positive y -intercepts. By plotting the more common isotope ( 86 Sr) against 87 Sr: 86 Sr ratio in Fig. 1 b, it is possible to visualize the consequences of the “invisible” differences in the 3 regressions from Fig. 1 a. In Fig. 1 b, the values at the far left represent the background ratios at the starting point. The differences reflected in 87 Sr: 86 Sr ratio (at the 3rd and 4th decimal place) in this artificial example are in the range of many 87 Sr: 86 Sr ratios reported. As source material is added (which has a lower 87 Sr: 86 Sr ratio than the existing system), values for the 87 Sr: 86 Sr ratio shown in Fig. 2 a asymptotically approach the values for the slopes of the regressions in Fig. 1 a. This occurs because for all linear functions y = m x + b, the ratio is defined by y / x = (1/ x ) b + m. For high values of 86 Sr or total Sr the (1/ x ) b term approaches zero, thus the ratio value approaches the original 86 Sr vs. 87 Sr slope. In Fig. 1 c, the existing system background has a lower 87 Sr: 86 Sr ratio than the material being added, resulting in three distinct functions with very small negative y -intercepts. The consequences of these differences are more easily visualized in Fig. 1 d, where the values at the far left represent the background ratios at the starting point. Note that these y -axis values are therefore generally lower than the values in Fig. 1 b. However, in both Fig. 1 b and d, the values at the far right are approaching the ratio of 0.7091 found in the imaginary modern seawater source. As source material with a higher 87 Sr: 86 Sr ratio than the background is added, the 87 Sr: 86 Sr ratio increases and asymptotically approaches the slope of the 86 Sr vs. 87 Sr regression lines in Fig. 1 c. Again, this occurs because for all linear functions y = m x + b; the ratio is defined by y / x = (1/ x ) b + m. For high values of 86 Sr or total Sr the (1/x) b term approaches zero, thus the ratio value, once again, approaches the original 86 Sr vs. 87 Sr slope. The relationship between the y -intercepts of the 86 Sr vs. 87 Sr artificial datasets (created to have a slope that is equivalent to modern seawater) from Fig. 1 a and c, plotted against the isotopic ratio of the system prior to any addition shows that y -intercepts are perfectly related to the backgrounds of the artificial examples (Fig. 1 e). The three blue points on the far right in Fig. 1 e represent the scenarios for positive y -intercepts shown in Fig. 1 a. The three blue points on the far left represent the scenarios for negative y -intercepts shown in Fig. 1 c. The central red point represents an initial 87 Sr: 86 Sr ratio which is the same as the exogenous additions. Under these conditions (source = background) the 87 Sr: 86 Sr ratio will not change, as is demonstrated in the red lines in Fig. 1 b and d. A plot of denominator vs. numerator ( y = m x + b) reveals both the source (m) and a value related to background (b) which, like the mathematically related mixing diagram, allows one to interpret isotopic data with respect to sources and backgrounds instead of relying only on observed ratios. In a review of 24 Sr isotope publications, with over 75 datasets, relationships between ratio denominators and numerators were linear for more than 66% of the cases (data not shown but available on request). In all cases where isotopic ratios changed with time, depth, or treatment, total Sr also changed. Non-zero y -intercepts for linear denominator isotope vs. numerator isotope concentration plots ( y = m x + b) are likely ubiquitous. There are several ways to correct ratio confounding errors related to non-zero y -intercepts in denominator vs. numerator plots. One approach is adjusting standard isotopic ratios (always calculated on a molar basis) to remove the background and total element effects that confound interpretation of isotope ratios. 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 and then a “new” calculated ratio calculated from the modified numerator and original denominator value. This will produce “corrected” isotopic ratios where the average of relatively similar ratio values matches the slope of denominator vs. numerator regression lines (Supplemental Material 8; and 16 ). The flat y -intercept adjusted mixing lines have the same y -intercept as a standard mixing line. Differences in corrected ratios that arise from less than perfect denominator vs. numerator relationships represent source changes for corresponding treatments, sampling time, or depth (Supplemental Material 8; and 16 ). For linear denominator vs. numerator plots, even with very high r 2 values, points do not fall exactly on the regression line. Points slightly above (+ residual) or below (- residual) the line respectively represent a source slightly higher or lower than the regression line slope. These residuals are strongly related to the y -intercept modified ratios (data not shown). The data in Fig. 1 a-d) could also be viewed as a system that is losing total element. In this case the 87 Sr: 86 Sr values at the far right of Fig. 1 c and d represent the background of systems before any Sr has been removed. Because the observed ratios in Fig. 1 b are greater than the loss ratio the system will become more enriched as higher 87 Sr: 86 Sr material is left behind. In Fig. 1 d the observed ratios are smaller than the loss ratio, thus the system will become more depleted as lower 87 Sr: 86 Sr material is left behind. The first scenario (Fig. 1 b) mimics isotopic fractionation as total Sr is lost from a system. Chemical or biological discrimination favoring the loss of the lighter isotope could lead to more enrichment of the heavy isotope. However, this increase in 87 Sr: 86 Sr enrichment is caused by a constant loss source that is independent of substrate levels (observed ratios) rather than constant discrimination (defined by a constant fractionation factor) that favours the light isotope. The y -intercepts in Fig. 1 a and c are also perfectly correlated with the before-loss backgrounds at the far right in Fig. 1 b and d. (data not shown). There is a much smaller range of y -axis values than what appears in Fig. 1 e, but the r 2 value is still 1.0. Denominator vs. numerator slopes for either the artificial (Fig. 1 a and c) or real data will define a net source (either loss or gain), but give no hint about whether a system is gaining, losing, or experiencing up and down changes in total element. The changing slopes of curvilinear functions described below have been viewed as representing a system with increasing total element to more simply describe the dynamics of observed ratios. However, these systems could also be viewed as losing total element. 2.2 Mathematical Relationships for Ratio-Based Expressions in Artificially Created Curvilinear Denominator vs. Numerator Relationships Additional artificial datasets were used to investigate curvilinear relationships. The first example was derived from data presented by Grupe et al. 3 in their evaluation of human bone and teeth. This process, using the best-fit relationship for real data, ensures that the curvatures are realistic. The umole/g units used by Grupe et al. 3 have been expressed as umole/ml to create a liquid example analogous to Fig. 1 . Concave functions with negative and positive y -intercepts were respectively created by subtracting or adding 0.05 moles/ml to the original polynomial predicted 87 Sr values. This creates three functions with similar curvature differing only in their y -intercepts. The 0.05 moles/ml adjustment is a small portion (~ 0.09%) of the original 87 Sr data range (~ 60 umoles/ml), but is sufficient to make an important difference in the 87 Sr: 86 Sr ratios because differences in the third and fourth decimal are commonly evaluated. Equation-derived functions (formulas in Supplemental Material Section 8; Note on adjusting ratios to account for non-zero y-intercepts for polynomial functions) that predict the isotopic ratios for a polynomial equation with a zero y -intercept exactly match the zero y -intercept corrected isotopic ratios. Additional comments on the importance of very small y -intercepts which will apply to either linear or curvilinear denominator vs. numerator functions are presented in Supplemental Material: Section 9; Note on the magnitude of non-zero y-intercepts. All three functions in Fig. 2 a have the same constantly declining derivative (not shown) as total Sr or 86 Sr increases, but only for the function with a zero y -intercept will derivatives and 87 Sr: 86 Sr ratios provide similar information. The concave polynomial relationship for an 86 Sr vs. 87 Sr relationship suggests that the source is becoming more depleted with added material. Evaluation of this new set of artificially created examples (Fig. 2 a) results in three possible patterns: (A) if the y -intercept is zero, the ratio of 87 Sr: 86 Sr will decline as more Sr is added to the system, simply because the source is becoming increasingly depleted. In this case, the original curvature (based on a changing source) is the only driver for changes in the isotopic ratio. (B) If the y -intercept is positive (indicating that the original source is less than the background), the ratio of 87 Sr: 86 Sr will decline as more source is added to the system. In this case, the decline in the ratio is most rapid initially where the ratio overrestimates what function curvature suggests. (C) Interestingly, if the y -intercept is negative, initially there is an increase in the ratio of 87 Sr: 86 Sr as the total Sr or 87 Sr increases. This occurs because the source has a higher ratio than the background. Ultimately as the source becomes more depleted, there is a decline in the 87 Sr: 86 Sr ratio of the system. Convex functions were created by reversing the sign of the x 2 term in the original relationship and then negative and positive y -intercepts were once again created by subtracting or adding 0.05 umoles/ml to the original 87 Sr values. This also creates three functions with similar curvature differing only in their y -intercepts. Note that a convex polynomial relationship defined by 86 Sr vs. 87 Sr plots suggests that the source is becoming more enriched with added material. Evaluation of these artificially created curvilinear datasets (Fig. 2 b) results in three different patterns than what are presented in Fig. 2 a: (A) if the y -intercept is zero, the 87 Sr: 86 Sr ratio will increase as more Sr is added to the system, simply because the source is more enriched than the system itself. In this case, the original 86 Sr vs. 87 Sr curvature (based on a changing source) is the only driver for changes in the isotopic ratio. (B) If the y -intercept is positive (indicating that the original source is less than the background), the ratio of 87 Sr: 86 Sr will initially decrease as more Sr is added to the system, but as the source becomes increasingly enriched, the ratio will begin to increase as the source becomes more influential than the background. (C) If the y -intercept is negative, the ratio of 87 Sr: 86 Sr increases as more enriched Sr is added to the system. In this case, the increase in the ratio is most rapid initially where the ratio underestimates what function curvature suggests. Supplemental Material; Section 10 simulations of curvilinear denominator vs. numerator relationships also suggest that the non-zero y -intercepts in polynomial functions are perfectly related to system backgrounds. 2.3 Quantifying Isotopic Sources An introduction of isotopic sources is presented in Supplemental Material: Section 5, Quantifying isotopic sources and Supplemental Material and Section 6, More nuances of Keeling plots and mixing diagrams). Here we include source evaluations of material presented in Fig. 1 a-e. A mixing diagram for the data presented in Fig. 1 a and c is presented in Fig. 3 a. For the 1/denominator (1/ 86 Sr in umole/ml units) form of the mixing diagrams, slopes match the y -intercepts shown in Fig. 1 a and c. The y -intercepts of the mixing diagrams match the slope of the original 86 Sr vs. 87 Sr plots shown in Fig. 1 a and c. As shown above, the y -intercepts of 86 Sr vs. 87 Sr plots are related to backgrounds. Either a mixing diagram (Fig. 3 a) or an 86 Sr vs. 87 Sr plot can identify exogenous sources and relative backgrounds. A different form of mixing diagram for the data in Fig. 1 a and c is presented in Fig. 3 b. For this example, the 1/total Sr concentration expression in umole/ml units is used as the x -axis rather than the 1/ 86 Sr (in umole/ml units) used in Fig. 3 a. For the 1/total element (in umole/ml units) form of the mixing diagram, the slopes do not exactly match the y -intercepts shown in Fig. 1 a and c. The mixing line slopes match denominator vs. numerator y -intercepts if the 1/denominator form of the mixing line is used because the two equations are equivalent y = m x + b and y / x = (1/ x ) b + m. The x terms differ for these two equations if the 1/total umoles element form (in Fig. 3 b) of the mixing line is used. The y -intercepts (in Fig. 3 a and b) of the mixing diagrams that represents a source for both forms match the slopes of the original 86 Sr vs. 87 Sr plots shown in Fig. 1 a and c. This occurs because the denominator of an isotopic ratio and total element are almost perfectly related because the portion of total for the denominator varies over a miniscule range. The 1/total element values in either ug/ml or umoles/ml units are a surrogate for 1/ 87 Sr in umole/ml units. Conventional mixing diagrams in the isotope literature usually use 1/total Sr concentrations in mass units (ug/ml in Fig. 3 c) rather than molar units that are shown in both Fig. 3 a and b). As stated above the y -intercept of the 1/total element concentration in mass units mixing line form also correctly identifies the source. Only the 1/ 86 Sr form (Fig. 3 a) of mixing line (using molar values) produces slopes exactly matching y -intercepts in Fig. 1 a and c. Different background isotopic composition can cause mathematical-driven differences in y -intercepts for denominator isotope vs. numerator isotope plots. However, just because y -intercepts differ, does not necessarily suggest that background differences are the sole cause. The y -intercepts of 86 Sr vs. 87 Sr regression equations are extremely small. Source detection, either from denominator isotope vs. numerator isotope slope; or indirectly from standard Keeling plots or mixing diagram y -intercepts, are less prone to error and can be more accurately calculated than background isotopic composition. Linear denominator isotope vs. numerator isotope functions ( y = m x + b) must produce linear mixing diagrams ( y / x = (1/ x ) b + m). Detectable curvature in a Keeling plot or mixing diagram suggests that the original denominator isotope vs. numerator isotope function is not linear. Therefore, an evaluation for possible Keeling plot or mixing line curvature can supplement denominator vs. numerator evaluations for statistically significant x 2 terms in polynomial best-fit equations, and spline smoothed cubic derivatives when assessing whether a denominator vs. numerator plot is linear. 2.4 Sources, Backgrounds and y-intercepts for Curvilinear Denominator vs. Numerator Functions The purpose of the simulations (presented in Supplemental Material: Section 10, Simulations demonstrating source, background and y -intercept effects on isotopic ratios for curvilinear denominator vs. numerator functions) presented here reiterate the fact that changing observed isotopic ratios are not necessarily related to changing sources. A second simulation goal is to demonstrate that interpreting a nonlinear mixing diagram, although difficult, can still be theoretically used for a changing source system, as data can be manipulated to deal with changing sources. A third goal was to demonstrate that even though y -intercepts for curvilinear 86 Sr vs. 87 Sr relationships in real datasets may be difficult to evaluate, they are mathematically related to system backgrounds. 2.5 Even if All Treatments in a Study are Defined with Linear Denominator vs. Numerator Functions, Treatments are not Necessarily Similarly Confounded. The size of the non-zero y -intercept and the distance of data points from the origin affect isotopic ratios’ confounding severity. The effects of different y -intercepts and the position of points (both close to and farther away from the origin) can be clearly seen in Fig. 1 b and d. The distance from the origin is important because at large values of the denominator a ratio will approach the denominator vs. numerator slope. For treatments with similar slopes and positive denominator vs. numerator y -intercepts, an average isotope ratio will be larger for a treatment with many low denominator values than for a treatment with higher denominator values. The Supplemental Material: Section 11, Simulation demonstrating why not all treatments in an experimental study are similarly confounded outlines how approximated quantitative numerical indicators of the degree of isotopic confounding can be theoretically defined. This is done by combining both y -intercept and denominator size effects. A similar real data example from Moyo et al. 25 reveals that 15 N: 14 N ratios are overestimated for all species in a trophic level study 16 Supplemental Material 2. While all six species have positive y -intercepts for linear 14 N vs. 15 N plots, scaling-related overestimates are severe for only two of six species evaluated. 3. Discussion Differences in isotopic ratio for various times, samples or treatments depend on three things for linear denominator vs. numerator relationships: First, the isotopic ratio of material being exogenously added to, or endogenously removed. Second, the initial isotopic ratio for the system to which material was exogenously added to, or endogenously removed from, and third, total element concentration, which is always strongly correlated to the denominator of an isotopic ratio. Factors ( 1 ) and ( 2 ) are respectively referred to as sources and backgrounds in isotopic studies 3,26 . Since ratio values increase and approach the denominator vs. numerator slope for linear functions with negative y -intercepts and decrease to approach the denominator vs. numerator slope for functions with positive y -intercepts, a ratio will almost always underestimate (for negative y -intercepts) or overestimate (for positive y -intercepts) the denominator vs. numerator slope. These under and overestimates of the denominator vs. numerator slope will likely create ratios and slopes that are not strongly related. Keeling plots and isotopic mixing diagrams are mathematically indirect ways to calculate the slope and relative y -intercepts of linear denominator vs. numerator concentration isotope plots. Linear non-zero y -intercepts are related to system backgrounds. Without knowing total element amounts, whether differing isotopic ratios suggest changing sources or simple dilution or concentration (as an observed ratio approaches the linear denominator vs. numerator slope as total element increases) cannot be determined. If the relationship between denominator isotope vs. numerator isotope is not linear, a fourth factor (denominator vs. numerator function curvature) also affects isotopic ratios, in ways (as shown above) that are intuitively difficult to predict. Summaries of the interpretive consequences of non-zero y -intercepts for both linear and curvilinear functions is presented in Supplemental Material: Section 12, Supplementary Table S1 and S2. Table S1 summarizes the mathematical consequences of likely ubiquitous non-zero, y -intercepts for isotopic ratio denominator vs. isotopic ratio numerator relationships. The diagrams indicate the shape of the denominator vs. observed ratio plot, and the shape of the denominator isotopic concentration vs. numerator isotope concentration plot for nine function categories. For an easier reference, Table S2 presents the same graphics in Table S1 but without the narrative. Isotopic discrimination can also lead to curvilinear denominator vs. numerator relationships. For example, if there is discrimination against a rarer heavier isotope, a substrate would become more enriched in the heavy isotope as it decomposes to a product. Either source changes or isotopic discrimination can create curvilinear relationships. Significant x 2 terms in denominator vs. numerator polynomial best-fit equations, and consistently changing spline smoothed cubic derivatives are useful for curvature detection regardless of its cause. Most importantly, changes in observed isotopic ratios do not necessarily reflect source changes. Furthermore, standard equations that convert isotopic ratios to the portion of total element that is derived from a source (such as N derived from fertilizer [NDFF]) do not account for the fact that isotopic ratios can be altered by total element concentration. Similarly, procedures to calculate discrimination factors 27 , if calculated from confounded isotopic ratios, are subject to error. Many variables scale with body size 28 . Therefore, isotopic ratios could depend on total element content, and thus complicate standard isotopic interpretations 12 . For another example, it is possible that massive differences in tree tissue sizes (flower buds, flowers, leaves, branches, roots, structural wood) imply that total N content for different tissues may differ over several orders of magnitude. These size differences could affect 15 N: 14 N ratios and the NDFF values derived from these ratios. Without accessing total N contents it may be difficult to determine if different isotopic ratios represent real differences in physiological partitioning or are an indirect scaling effect. We emphasize that not all isotopic datasets are confounded. Patterns where a source changes with exogenous gains or endogenous losses can also be much more complex than the relatively simple constant source (linear) and consistently-changing source (polynomial) models illustrated here. However, one will never know if isotopic data is confounded unless it is evaluated. Isotopic researchers need encouragement to change. Funding agencies should not support isotopic research unless proposed studies include source analyses. 4. Methods Since this first paper in a two-paper series deals with theoretical mathematical relationships that will be associated with misinterpretation in isotopic data, detailed methods are not included. All artificial data produced here have realistic data ranges and reflect what will be demonstrated in the companion publication 16 with real data. Sr isotopes were used to demonstrate mathematical principles because 87 Sr: 86 Sr ratios are less confusing for many statistically -oriented readers unfamiliar with the delta notation used for other common isotopes. Declarations Additional Informationl Additional Information is included in Supplemental Material 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 the first draft of the manuscript once figure legends and tables were created by TLR, CPN and CN. 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. Acknowledgements Sincere thanks and love to TLR’s life Mary Ann Righetti (BS,MLS,JD). She has been a sounding board for the many years of concept development. Her non-STEM related evaluations improve clarity. Furthermore, she is the originator of the ideas and format behind Table 2, which goes a long way to making the manuscript comprehensible. Mary Ann also encouraged us to include the Martian data. 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 Statement Upon acceptance all data that is presented in Figures or Tables in both the main text or supplemental material will either be posted with a general-purpose archival service such as Dryad or included in supplementary material. References Banner, J. L., Musgrove, M., Asmerom, Y., Edwards, R. L. & Hoff, J. A. High-resolution temporal record of Holocene ground-water chemistry: tracing links between climate and hydrology. Geology 24, 1049–1053 (1996). Ericson, J. E. Strontium isotope characterization in the study of prehistoric human ecology. J. Hum. Evol. 14, 503–514 (1985). Grupe, G. et al. Mobility of bell beaker people revealed by strontium isotope ratios of tooth and bone: a study of southern bavarian skeletal remains. Appl. Geochem. 12, 517–525 (1997). Gannes, L. Z., del Rio, C. M. & Koch, P. Natural abundance variations in stable isotopes and their potential uses in animal physiological ecology. Comp. Biochem. Physiol. A Mol. Integr. Physiol. 119, 725–737 (1998). Ehleringer, J. R., Hall, A. E. & Farquhar, G. D. Stable isotopes and plant carbon-water relations (Academic Press, 1993). Jasienski, M. & Bazzaz, F. A. The fallacy of ratios and the testability of models in biology. Oikos 84, 321–326 (1999). Phillips, D. L., and Koch, P. L. “Incorporating Concentration Dependence in Stable Isotope Mixing Models,” Ecologies, 130, 114–125, (2002), Pearson, S. F., Levey, D. J., and Greenberg, C. H., “Effects of Elemental Composition on the Incorporation of Dietary Nitrogen and Carbon Isotopic Signatures in an Omnivorous Songbird,” Oecologia, 135, 516–523 (2003). Righetti, T. L., Dalthorp, D., Sandrock, D., Strik, B., Banados, P., and Zhou, Z. “Slope-Based and Ratio-Based Approaches to Determine Fertiliser-Derived N in Plant Tissues for Established Perennial Plants,” The Journal of Horticultural Science and Biotechnology, 82, 641–647 (2007b). Righetti, T.L., Sandrock, D.R., Strik, B. and Azarenko, A. “Appropriate analysis and interpretation approaches to determine fertilizer-derived nitrogen in plant tissues.” Journal of the American Society for Horticultural Science, 132, 429–436, (2007c Righetti, T. L., Dalthorp D., Lambrinos, J. D., Strik, B., Sandrock, D., and Phillips, C. “Scaling Confounds the Interpretation of Isotopic Data,” International Journal of Environmental Analytical Chemistry, 92, 1–27 (2012). Tejada-Lara JV, MacFadden BJ, Bermudez L, Rojas G, Salas-Gismondi R, Flynn JJ. 2018 Body mass predicts isotope enrichment in herbivorous mammals. Proc. R. Soc. B 285: 2018 – 1020 (2018). Villamarín, F., Jardine, T. D., Bunn, S. E., Marioni, B., and Magnusson, W. E. “Body Size is More Important Than Diet in Determining Stable-Isotope Estimates of Trophic Position in Crocodilians,” Scientific Reports, 8, 1–11 (2018). Jacobi, C. M., Villamarin, F., Jardine, T. D., and Magnusson, W.E. “Uncertainities Associated with Trophic Discrimination Factor and Body Size Complicate Calculation of d 15 N-Derived Trophic Positions in Arapaima sp.,” Ecology of Freshwater Fish, 29, 779–789 (2020). , Whitledge, G. W., and Rabeni, C. F. “Energy Sources and Ecological Role of Crayfishes in an Ozark Stream: Insights from Stable Isotopes and Gut Analysis,” Canadian Journal of Fisheries and Aquatic Sciences, 54, 2555–2563 (1997). 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: dealing with confounded isotopic ratios. Scientific Reports this issue (2024). 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). Righetti, T. L. et al. Analysis of ratio-based responses. J. Am. Soc. Hortic. Sci. 132, 3–13 (2007). 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). Sagan, C. Encyclopaedia galactica in Cosmos: a personal voyage. Episode 12 24 (PBS, 1980). Keeling, C. D. The concentration and isotopic abundances of atmospheric carbon dioxide in rural areas. Geochim. Cosmochim. Acta 13, 322–334 (1958). Pataki, D. E. et al. The application and interpretation of Keeling plots in terrestrial carbon cycle research. Glob. Biogeochem. Cycles 17, 1022 (2003). Goede, A., McCulloch, M., McDermott, F. & Hawkesworth, C. Aeolian contribution to strontium and strontium isotope variations in a Tasmanian speleothem. Holocene 9, 715–722 (1998). 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). Bilby, R. E., Fransen, B. R. & Bisson, P. A. Incorporation of nitrogen and carbon from spawning coho salmon into the trophic system of small streams: evidence from stable isotopes. Can. J. Fish. Aquat. Sci. 53, 164–173 (1996). Mariotti, A. et al. Experimental determination of nitrogen kinetic isotope fractionation: some principles; illustration for the denitrification and nitrification processes. Plant Soil 62, 413–430 (1981). Savage, V. M. et al. The predominance of quarter-power scaling in biology. Funct. Ecol. 18, 257–282 (2004). Additional Declarations No competing interests reported. Supplementary Files 2TheorySupplementaltlrmarch2.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. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-4086468","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":283467124,"identity":"39d62e58-2535-457e-914a-18a6d4496bd9","order_by":0,"name":"Kate Moots","email":"","orcid":"","institution":"University of Guam","correspondingAuthor":false,"prefix":"","firstName":"Kate","middleName":"","lastName":"Moots","suffix":""},{"id":283467125,"identity":"318422e1-a965-4ba5-a3f2-4678205bfe75","order_by":1,"name":"Christina P. Nguyen","email":"","orcid":"","institution":"University of Guam","correspondingAuthor":false,"prefix":"","firstName":"Christina","middleName":"P.","lastName":"Nguyen","suffix":""},{"id":283467126,"identity":"8d1e39b7-8bb8-443c-90df-62094d6a0cdf","order_by":2,"name":"Catherine Nguyen","email":"","orcid":"","institution":"University of Guam","correspondingAuthor":false,"prefix":"","firstName":"Catherine","middleName":"","lastName":"Nguyen","suffix":""},{"id":283467127,"identity":"ed7acb09-7899-4f5b-a9e6-250b990c54bb","order_by":3,"name":"Frank Camacho","email":"","orcid":"","institution":"University of Guam","correspondingAuthor":false,"prefix":"","firstName":"Frank","middleName":"","lastName":"Camacho","suffix":""},{"id":283467128,"identity":"9261ba30-c428-4714-8418-ae5294014bdc","order_by":4,"name":"Dan Lindstrom","email":"","orcid":"","institution":"University of Guam","correspondingAuthor":false,"prefix":"","firstName":"Dan","middleName":"","lastName":"Lindstrom","suffix":""},{"id":283467129,"identity":"2c8d741b-1a79-4324-99a4-4ea7825e432c","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 16:56:01","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4086468/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4086468/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":53449931,"identity":"7b87c2be-7d54-43dd-8b81-d48396da8d36","added_by":"auto","created_at":"2024-03-26 06:27:47","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":840176,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eLinear artificial Sr isotope relaltionships that mimic real data\u003c/strong\u003e \u003cstrong\u003ea\u003c/strong\u003e. Artificial \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr data created to have a slope that is equivalent to the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio of modern seawater. Functions with very small positive \u003cem\u003ey\u003c/em\u003e-intercepts have been created to demonstrate the consequences of very small deviations from an isometric relationship. Although the three different regression lines are so similar that they appear as one line, there are three different regression equations. The positive \u003cem\u003ey\u003c/em\u003e-intercepts represent 0.01, 0.02 and 0.03 percent of the \u003csup\u003e87\u003c/sup\u003eSr range (~10 ppm). Although very small these intercepts still lead to interpretive problems as shown in Fig. 1b. Estimates for total Sr based on a constant assumption for the portion of \u003csup\u003e86\u003c/sup\u003eSr are also shown in brackets on the \u003cem\u003ex\u003c/em\u003e-axis; \u003cstrong\u003eb.\u003c/strong\u003e Artificial \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr data created to have a slope that is equivalent to the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio of modern seawater and small positive \u003cem\u003ey\u003c/em\u003e-intercepts shown in Fig. 1a are replotted here. The \u003csup\u003e86\u003c/sup\u003eSr concentrations are plotted against \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios for each of the three functions. The initial values (backgrounds) for each of the three functions (far left points) represent the starting point of the systems before any Sr-containing material was added. Differences reflected in \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio in this artificial example are important. The differences in \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios are realistic, and in the range of many \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios reported in the literature. Estimates for total Sr based on a constant assumption for the portion of \u003csup\u003e86\u003c/sup\u003eSr are also shown in brackets on the \u003cem\u003ex\u003c/em\u003e-axis; \u003cstrong\u003ec.\u003c/strong\u003e Artificial \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr data created to have a slope that is equivalent to the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio of modern seawater. Functions with very small negative \u003cem\u003ey\u003c/em\u003e-intercepts have been created to demonstrate the consequences of very small deviations from an isometric relationship. The negative \u003cem\u003ey\u003c/em\u003e-intercepts represent 0.01, 0.02 and 0.03 percent of the \u003csup\u003e87\u003c/sup\u003eSr range (~10 ppm). Although the three different regression lines are so similar that they appear as one line, there are in fact three different regression equations. Estimates for total Sr based on a constant assumption for the portion of \u003csup\u003e86\u003c/sup\u003eSr are also shown in parenthesis on the \u003cem\u003ex\u003c/em\u003e-axis; \u003cstrong\u003ed. \u003c/strong\u003eArtificial \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr data created to have a slope that is equivalent to the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio of modern seawater and small negative \u003cem\u003ey\u003c/em\u003e-intercepts shown in Fig. 1c are replotted here. The \u003csup\u003e86\u003c/sup\u003eSr concentrations are plotted against \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios for each of the three functions. The initial values (backgrounds) for each of the three functions (far left points) represent the starting point of the systems before any Sr-containing material was added. Values for \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio asymptotically approach the slope of the \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr regression lines (Fig. 1c) for high values of either \u003csup\u003e86\u003c/sup\u003eSr or total Sr concentrations. The differences in \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios shown here are in the range of many \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios reported in the literature. Estimates for total Sr based on a constant assumption for the portion of \u003csup\u003e86\u003c/sup\u003eSr are also shown in brackets on the \u003cem\u003ex\u003c/em\u003e-axis; \u003cstrong\u003ee.\u003c/strong\u003e The relationship between the \u003cem\u003ey\u003c/em\u003e-intercepts of artificial \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr data created to have a slope that is equivalent to the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio of modern seawater (shown in Fig. 1a and c) and the isotopic ratio of the system before the addition of exogenous Sr (the backgrounds). Backgrounds are identified as the initial \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios at the far left in Fig. 1b and d. The initial \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio of the system (referred to as background in most isotopic literature) is perfectly related to the \u003cem\u003ey\u003c/em\u003e-intercepts.\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-4086468/v1/a646b08456374d61da634d19.png"},{"id":53449929,"identity":"4b6e98be-38ca-43ff-9cdc-744dfdefdb42","added_by":"auto","created_at":"2024-03-26 06:27:47","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":267839,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eCurvilinear artificial Sr isotope relaltionships that mimic real data\u003c/strong\u003e \u003cstrong\u003ea.\u003c/strong\u003e The relationship between \u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios for three perfect concave polynomial relationships with negative, zero and positive y-intercepts (respectively from bottom to top). All three functions have the same derivative (constant declining function), but only for a function with a zero y-intercept, will derivatives and \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios provide similar information; \u003cstrong\u003eb.\u003c/strong\u003e The relationship between \u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios for three perfect convex polynomial relationships with negative, zero and positive \u003cem\u003ey\u003c/em\u003e-intercepts (respectively from bottom to top). All three functions have the same derivative (constant declining function), but only for a function with a zero \u003cem\u003ey\u003c/em\u003e-intercept will derivatives and \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios provide similar information.\u003c/p\u003e\n\u003cp\u003eAll panels are available as individual files in *.pdf format.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4086468/v1/dc8b43c146f6c26f1895914d.png"},{"id":53450603,"identity":"dde9407f-0dd7-4c4e-a17a-79e6f1a3e31a","added_by":"auto","created_at":"2024-03-26 06:35:49","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":533554,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eArtificial source analyses for Sr isotopes\u003c/strong\u003e \u003cstrong\u003ea.\u003c/strong\u003e Graphical illustration of the Keeling plot method (referred to as a mixing diagram for non-gas samples) for the artificial data presented in Fig. 1a and c. The expression 1/\u003csup\u003e86\u003c/sup\u003eSr (in umoles/ml) rather than the standard approach using total element (total Sr in mass units) is presented on the \u003cem\u003ex\u003c/em\u003e-axis; \u003cstrong\u003eb.\u003c/strong\u003e Graphical illustration of the Keeling plot method (referred to as a mixing diagram for non-gas samples) using total element (total Sr in umole/ml units) on the \u003cem\u003ey\u003c/em\u003e-axis for the artificial data presented in Fig. 1a and c; \u003cstrong\u003ec.\u003c/strong\u003e Graphical illustration of the Keeling plot method (referred to as a mixing diagram for non-gas samples) using the conventional form with total element concentration in mass units. This figure is similar to Fig. 3b except the units on the \u003cem\u003ex\u003c/em\u003e-axis for total Sr are in ug/ml rather than in umole/ml. This form of mixing diagram (1/total Sr based on mass units (ug/ml) vs. \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr observed ratios (umole basis) is the standard form used in the isotopic literature. Again, this figure is calculated from the same artificial data presented in Fig. 1a and c;\u003c/p\u003e\n\u003cp\u003eAll panels are available as individual files in *.pdf format.\u0026nbsp;\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4086468/v1/7d13a70033433ecff79743d8.png"},{"id":56909775,"identity":"5e12ebc9-f008-4759-a2f5-fdd3cf7001e8","added_by":"auto","created_at":"2024-05-22 04:37:30","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2173439,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4086468/v1/30b5fea8-c1b4-4c53-ae96-4fe3ea6fae3b.pdf"},{"id":53449932,"identity":"d497ab63-8c07-4b06-aeb9-92c28c5c702b","added_by":"auto","created_at":"2024-03-26 06:27:47","extension":"docx","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":1331236,"visible":true,"origin":"","legend":"","description":"","filename":"2TheorySupplementaltlrmarch2.docx","url":"https://assets-eu.researchsquare.com/files/rs-4086468/v1/a24f5475b8acb969f7b3e1e0.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Mathematical Explanation for Why Ratio-Based Isotopic Analyses are Commonly Misleading: Theory","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eObserved stable mass isotopes serve as a proxy for past climates\u003csup\u003e1\u003c/sup\u003e and are used as tracers in archaeological\u003csup\u003e2,3\u003c/sup\u003e, ecological\u003csup\u003e4\u003c/sup\u003e, geological\u003csup\u003e1\u003c/sup\u003e and plant or animal physiological studies\u003csup\u003e,5\u003c/sup\u003e (see Supplemental Material: Section \u003cspan refid=\"Sec1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, Review of isotopic principles). These applications involve evaluations of observed isotopic ratios or ratio-based expressions (see Supplemental Material: Section \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, Formulas for converting isotopic ratios to ratio-based expressions). Values of variables calculated from observed ratios are also similarly evaluated (see Supplemental Material: Section \u003cspan refid=\"Sec7\" class=\"InternalRef\"\u003e3\u003c/span\u003e, Formulas for converting isotopic ratios to portion of total element derived from an added source) for an example.\u003c/p\u003e \u003cp\u003eThis mathematical and statistical report emphasizes source (isotopic ratio of material being added or lost) analyses and challenges current methods for evaluating isotopes. Isotopic studies usually involve standard statistical evaluations of observed isotopic ratios or ratio-based expressions for different times, treatments, or conditions. These standard procedures have flourished despite warnings about using ratios in research studies being repeatedly reported for over a century\u003csup\u003e6\u003c/sup\u003e. Concerns that ratios or ratio-based isotopic expressions produce conclusions that do not support known physiological or ecological principles have been reported\u003csup\u003e7\u0026ndash;15\u003c/sup\u003e. Twelve additional examples of isotopic evidence that contradict other known facts are presented in this two-paper series. The project leader (TLR) has also made interpretive mistakes. A re-evaluation of the first dataset that caught his attention is discussed\u003csup\u003e16\u003c/sup\u003e (see Supplementary Material 3 in reference 16). A review of the ratio literature on which this challenge is based is presented (see Supplemental Material: Section \u003cspan refid=\"Sec8\" class=\"InternalRef\"\u003e4\u003c/span\u003e, Review of ratio literature and isotopic warnings).\u003c/p\u003e \u003cp\u003eWe emphasize that observed isotopic ratios for a system will depend on sources, backgrounds (isotopic ratio of a system before any loses or gains) and total element concentration or content differences among different times, treatments, or conditions. Concepts demonstrated here are important because ratios are often misleading as they depend on denominator size\u003csup\u003e6,17\u0026ndash;20\u003c/sup\u003e. Without measuring and analysing total element one cannot determine if changing observed isotopic ratios are caused by source changes or differing amounts of total element. We propose that source is the most important variable to analyse but, it is often ignored as statistical evaluations are mostly conducted on observed ratios.\u003c/p\u003e \u003cp\u003eOur presentation may qualify as an extraordinary claim. How could research methodologies associated with scientific breakthroughs in many scientific disciplines have mathematical flaws? We are pursuing an \u0026ldquo;extraordinary claims require extraordinary evidence\u0026rdquo; philosophy championed by Sagan\u003csup\u003e21\u003c/sup\u003e. More mathematical descriptions, artificial examples, simulations and data re-evaluations are presented in the text and supplemental material for this two-paper series than what is usually presented. Studies as diverse as \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr evaluations assessing climate change from cave speleothems\u003csup\u003e16\u003c/sup\u003e; \u003csup\u003e15\u003c/sup\u003eN:\u003csup\u003e14\u003c/sup\u003eN trophic level studies and \u003csup\u003e13\u003c/sup\u003eC:\u003csup\u003e12\u003c/sup\u003eC investigations focused on the possibility of previous life on Mars\u003csup\u003e16\u003c/sup\u003e (see Supplemental Material 2 in reference 16) suggest that source (loss or gain) evaluations and traditional approaches evaluating either isotopic ratios or ratio-based expressions produce very different interpretations. Current concepts related to isotopic fractionation are also challenged\u003csup\u003e16\u003c/sup\u003e (see Supplemental Materials 3 in reference 16). Our goal is to advocate better methods when analyzing isotopic ratios in the thousands of mass isotope publications annually produced. Evaluating sources, which requires knowledge of total element concentration, provides more detail than solely evaluating observed isotopic ratios.\u003c/p\u003e \u003cp\u003eKeeling\u003csup\u003e22\u003c/sup\u003e plots are commonly used to determine sources in gaseous \u003csup\u003e13\u003c/sup\u003eC studies\u003csup\u003e23\u003c/sup\u003e. Variations of Keeling\u0026rsquo;s original procedure referred to as mixing diagrams for non-gas samples are also used\u003csup\u003e24\u003c/sup\u003e. Details of these procedures are presented in supplementary material (see Supplemental Material: Section 5, Quantifying isotopic sources and Supplemental Material: Section 6, More nuances of Keeling plots and mixing diagrams).\u003c/p\u003e \u003cp\u003eA summary of 300 evaluations (100 each for \u003csup\u003e13\u003c/sup\u003eC, \u003csup\u003e15\u003c/sup\u003eN and \u003csup\u003e87\u003c/sup\u003eSr studies) shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e (see Supplemental Material: Section 7 for methods) demonstrates the magnitude of the problem. In our limited sample of gaseous carbon studies (6 of 100 \u003csup\u003e13\u003c/sup\u003eC publications), mixing diagrams are used in half of the combined CO\u003csub\u003e2\u003c/sub\u003e and CH\u003csub\u003e4\u003c/sub\u003e isotopic research publications. Otherwise, sixty years after being introduced for gaseous studies, the use of Keeling\u0026rsquo;s approach is rare. Only 17% of Sr isotopic publications use source analyses. Although common for CO\u003csub\u003e2\u003c/sub\u003e and CH\u003csub\u003e4\u003c/sub\u003e research, less than 10% of all isotopic C studies use the technique. For N isotopic research publications none used source analyses. The total element concentrations, necessary for source analyses, are likely available in less than half of isotopic publications.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePortion of \u003csup\u003e15\u003c/sup\u003eN, \u003csup\u003e13\u003c/sup\u003eC, and \u003csup\u003e87\u003c/sup\u003eSr publications that include information related to total element concentrations that can be matched with isotopic ratio data. Also included are the portion of total publications where a source analysis (Keeling plots or mixing diagrams) was utilized.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eTotal element data not included %\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c4\" namest=\"c3\"\u003e \u003cp\u003eTotal element data used to calculate sources\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e% of all publications\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e% of publications with total element\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eNon-gas publications\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003csup\u003e\u003cb\u003e15\u003c/b\u003e\u003c/sup\u003e\u003cb\u003eN\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003csup\u003e\u003cb\u003e13\u003c/b\u003e\u003c/sup\u003e\u003cb\u003eC\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e79\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e14\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003csup\u003e\u003cb\u003e87\u003c/b\u003e\u003c/sup\u003e\u003cb\u003eSr\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eGas publications\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003csup\u003e\u003cb\u003e15\u003c/b\u003e\u003c/sup\u003e\u003cb\u003eN\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e61\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003csup\u003e\u003cb\u003e13\u003c/b\u003e\u003c/sup\u003e\u003cb\u003eC\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e33\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e75\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eAll publications\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003csup\u003e\u003cb\u003e15\u003c/b\u003e\u003c/sup\u003e\u003cb\u003eN\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e65\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003csup\u003e\u003cb\u003e13\u003c/b\u003e\u003c/sup\u003e\u003cb\u003eC\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e70\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003csup\u003e\u003cb\u003e87\u003c/b\u003e\u003c/sup\u003e\u003cb\u003eSr\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e17\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e30\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e"},{"header":"2. Results","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Mathematical Relationships for Ratio-Based Expressions in Artificially Created Linear Denominator vs. Numerator Relationships\u003c/h2\u003e \u003cp\u003eThe first artificial datasets (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea-e) demonstrate the effect of very small differences in \u003cem\u003ey\u003c/em\u003e-intercepts for linear denominator vs. numerator functions on observed \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios. Plots of the umole/ml denominator vs. the umole/ml numerator of isotopic ratios as presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea and c are not evaluated in the isotopic literature. They are presented here to demonstrate the mathematical relationships between linear denominator vs. numerator plots and plots of the denominator or total isotope vs. an isotopic ratio. The datasets presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea-e, represent a system where Sr-containing material with a constant \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio is being added to or removed from an existing system, where the slope value for an \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr plot is equal to the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio of modern seawater. For simplicity, this artificial data (and similar data in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ec and d) is first imagined as coming from a system where Sr-containing material with a constant \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio is being added to an existing system. Slopes of denominator vs. numerator regression equations will therefore be related to sources (the isotopic ratio of an element gained). The same system viewed as losing total element will be subsequently discussed.\u003c/p\u003e \u003cp\u003eWith the linear equations as a starting point, it is easier to take the next step and demonstrate that isotopic ratios must depend on their denominators and total isotope concentrations (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003eb and d). The slope of the \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr regression line is the isotopic ratio of the material being added to the system. This dataset results in a linear relationship when \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr are plotted because the ratio of \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr in the source remains constant. In Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea, the existing system background has a higher \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio than the material being added. Small deviations from an isometric relationship result in 3 infinitesimally different regression lines, each with very small positive \u003cem\u003ey\u003c/em\u003e-intercepts. By plotting the more common isotope (\u003csup\u003e86\u003c/sup\u003eSr) against \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003eb, it is possible to visualize the consequences of the \u0026ldquo;invisible\u0026rdquo; differences in the 3 regressions from Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea. In Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003eb, the values at the far left represent the background ratios at the starting point. The differences reflected in \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio (at the 3rd and 4th decimal place) in this artificial example are in the range of many \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios reported.\u003c/p\u003e \u003cp\u003eAs source material is added (which has a lower \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio than the existing system), values for the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e2\u003c/span\u003ea asymptotically approach the values for the slopes of the regressions in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea. This occurs because for all linear functions \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;m\u003cem\u003ex\u003c/em\u003e\u0026thinsp;+\u0026thinsp;b, the ratio is defined by \u003cem\u003ey\u003c/em\u003e/\u003cem\u003ex\u003c/em\u003e = (1/\u003cem\u003ex\u003c/em\u003e) b\u0026thinsp;+\u0026thinsp;m. For high values of \u003csup\u003e86\u003c/sup\u003eSr or total Sr the (1/\u003cem\u003ex\u003c/em\u003e) b term approaches zero, thus the ratio value approaches the original \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr slope.\u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ec, the existing system background has a lower \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio than the material being added, resulting in three distinct functions with very small negative \u003cem\u003ey\u003c/em\u003e-intercepts. The consequences of these differences are more easily visualized in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ed, where the values at the far left represent the background ratios at the starting point. Note that these \u003cem\u003ey\u003c/em\u003e-axis values are therefore generally lower than the values in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003eb. However, in both Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003eb and d, the values at the far right are approaching the ratio of 0.7091 found in the imaginary modern seawater source. As source material with a higher \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio than the background is added, the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio increases and asymptotically approaches the slope of the \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr regression lines in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ec. Again, this occurs because for all linear functions \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;m\u003cem\u003ex\u003c/em\u003e\u0026thinsp;+\u0026thinsp;b; the ratio is defined by \u003cem\u003ey\u003c/em\u003e/\u003cem\u003ex\u003c/em\u003e = (1/\u003cem\u003ex\u003c/em\u003e) b\u0026thinsp;+\u0026thinsp;m. For high values of \u003csup\u003e86\u003c/sup\u003eSr or total Sr the (1/x) b term approaches zero, thus the ratio value, once again, approaches the original \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr slope.\u003c/p\u003e \u003cp\u003eThe relationship between the \u003cem\u003ey\u003c/em\u003e-intercepts of the \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr artificial datasets (created to have a slope that is equivalent to modern seawater) from Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea and c, plotted against the isotopic ratio of the system prior to any addition shows that \u003cem\u003ey\u003c/em\u003e-intercepts are perfectly related to the backgrounds of the artificial examples (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ee). The three blue points on the far right in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ee represent the scenarios for positive \u003cem\u003ey\u003c/em\u003e-intercepts shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea. The three blue points on the far left represent the scenarios for negative \u003cem\u003ey\u003c/em\u003e-intercepts shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ec. The central red point represents an initial \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio which is the same as the exogenous additions. Under these conditions (source\u0026thinsp;=\u0026thinsp;background) the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio will not change, as is demonstrated in the red lines in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003eb and d. A plot of denominator vs. numerator (\u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;m\u003cem\u003ex\u003c/em\u003e\u0026thinsp;+\u0026thinsp;b) reveals both the source (m) and a value related to background (b) which, like the mathematically related mixing diagram, allows one to interpret isotopic data with respect to sources and backgrounds instead of relying only on observed ratios.\u003c/p\u003e \u003cp\u003eIn a review of 24 Sr isotope publications, with over 75 datasets, relationships between ratio denominators and numerators were linear for more than 66% of the cases (data not shown but available on request). In all cases where isotopic ratios changed with time, depth, or treatment, total Sr also changed. Non-zero \u003cem\u003ey\u003c/em\u003e-intercepts for linear denominator isotope vs. numerator isotope concentration plots (\u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;m\u003cem\u003ex\u003c/em\u003e\u0026thinsp;+\u0026thinsp;b) are likely ubiquitous.\u003c/p\u003e \u003cp\u003eThere are several ways to correct ratio confounding errors related to non-zero \u003cem\u003ey\u003c/em\u003e-intercepts in denominator vs. numerator plots. One approach is adjusting standard isotopic ratios (always calculated on a molar basis) to remove the background and total element effects that confound interpretation of isotope ratios. 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 \u0026ldquo;new\u0026rdquo; 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 and then a \u0026ldquo;new\u0026rdquo; calculated ratio calculated from the modified numerator and original denominator value.\u003c/p\u003e \u003cp\u003eThis will produce \u0026ldquo;corrected\u0026rdquo; isotopic ratios where the average of relatively similar ratio values matches the slope of denominator vs. numerator regression lines (Supplemental Material 8; and\u003csup\u003e16\u003c/sup\u003e). The flat \u003cem\u003ey\u003c/em\u003e-intercept adjusted mixing lines have the same \u003cem\u003ey\u003c/em\u003e-intercept as a standard mixing line. Differences in corrected ratios that arise from less than perfect denominator vs. numerator relationships represent source changes for corresponding treatments, sampling time, or depth (Supplemental Material 8; and\u003csup\u003e16\u003c/sup\u003e).\u003c/p\u003e \u003cp\u003eFor linear denominator vs. numerator plots, even with very high r\u003csup\u003e2\u003c/sup\u003e values, points do not fall exactly on the regression line. Points slightly above (+\u0026thinsp;residual) or below (- residual) the line respectively represent a source slightly higher or lower than the regression line slope. These residuals are strongly related to the \u003cem\u003ey\u003c/em\u003e-intercept modified ratios (data not shown).\u003c/p\u003e \u003cp\u003eThe data in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea-d) could also be viewed as a system that is losing total element. In this case the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr values at the far right of Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ec and d represent the background of systems before any Sr has been removed. Because the observed ratios in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003eb are greater than the loss ratio the system will become more enriched as higher \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr material is left behind. In Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ed the observed ratios are smaller than the loss ratio, thus the system will become more depleted as lower \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr material is left behind.\u003c/p\u003e \u003cp\u003eThe first scenario (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003eb) mimics isotopic fractionation as total Sr is lost from a system. Chemical or biological discrimination favoring the loss of the lighter isotope could lead to more enrichment of the heavy isotope. However, this increase in \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr enrichment is caused by a constant loss source that is independent of substrate levels (observed ratios) rather than constant discrimination (defined by a constant fractionation factor) that favours the light isotope. The \u003cem\u003ey\u003c/em\u003e-intercepts in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea and c are also perfectly correlated with the before-loss backgrounds at the far right in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003eb and d. (data not shown). There is a much smaller range of \u003cem\u003ey\u003c/em\u003e-axis values than what appears in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ee, but the r\u003csup\u003e2\u003c/sup\u003e value is still 1.0.\u003c/p\u003e \u003cp\u003eDenominator vs. numerator slopes for either the artificial (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea and c) or real data will define a net source (either loss or gain), but give no hint about whether a system is gaining, losing, or experiencing up and down changes in total element. The changing slopes of curvilinear functions described below have been viewed as representing a system with increasing total element to more simply describe the dynamics of observed ratios. However, these systems could also be viewed as losing total element.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Mathematical Relationships for Ratio-Based Expressions in Artificially Created Curvilinear Denominator vs. Numerator Relationships\u003c/h2\u003e \u003cp\u003eAdditional artificial datasets were used to investigate curvilinear relationships. The first example was derived from data presented by Grupe \u003cem\u003eet al.\u003c/em\u003e\u003csup\u003e3\u003c/sup\u003e in their evaluation of human bone and teeth. This process, using the best-fit relationship for real data, ensures that the curvatures are realistic. The umole/g units used by Grupe \u003cem\u003eet al.\u003c/em\u003e\u003csup\u003e3\u003c/sup\u003e have been expressed as umole/ml to create a liquid example analogous to Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eConcave functions with negative and positive \u003cem\u003ey\u003c/em\u003e-intercepts were respectively created by subtracting or adding 0.05 moles/ml to the original polynomial predicted \u003csup\u003e87\u003c/sup\u003eSr values. This creates three functions with similar curvature differing only in their \u003cem\u003ey\u003c/em\u003e-intercepts. The 0.05 moles/ml adjustment is a small portion (~\u0026thinsp;0.09%) of the original \u003csup\u003e87\u003c/sup\u003eSr data range (~\u0026thinsp;60 umoles/ml), but is sufficient to make an important difference in the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios because differences in the third and fourth decimal are commonly evaluated. Equation-derived functions (formulas in Supplemental Material Section 8; Note on adjusting ratios to account for non-zero y-intercepts for polynomial functions) that predict the isotopic ratios for a polynomial equation with a zero \u003cem\u003ey\u003c/em\u003e-intercept exactly match the zero \u003cem\u003ey\u003c/em\u003e-intercept corrected isotopic ratios. Additional comments on the importance of very small \u003cem\u003ey\u003c/em\u003e-intercepts which will apply to either linear or curvilinear denominator vs. numerator functions are presented in Supplemental Material: Section 9; Note on the magnitude of non-zero y-intercepts.\u003c/p\u003e \u003cp\u003eAll three functions in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e2\u003c/span\u003ea have the same constantly declining derivative (not shown) as total Sr or \u003csup\u003e86\u003c/sup\u003eSr increases, but only for the function with a zero \u003cem\u003ey\u003c/em\u003e-intercept will derivatives and \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios provide similar information. The concave polynomial relationship for an \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr relationship suggests that the source is becoming more depleted with added material. Evaluation of this new set of artificially created examples (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e2\u003c/span\u003ea) results in three possible patterns: (A) if the \u003cem\u003ey\u003c/em\u003e-intercept is zero, the ratio of \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr will decline as more Sr is added to the system, simply because the source is becoming increasingly depleted. In this case, the original curvature (based on a changing source) is the only driver for changes in the isotopic ratio. (B) If the \u003cem\u003ey\u003c/em\u003e-intercept is positive (indicating that the original source is less than the background), the ratio of \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr will decline as more source is added to the system. In this case, the decline in the ratio is most rapid initially where the ratio overrestimates what function curvature suggests. (C) Interestingly, if the \u003cem\u003ey\u003c/em\u003e-intercept is negative, initially there is an increase in the ratio of \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr as the total Sr or \u003csup\u003e87\u003c/sup\u003eSr increases. This occurs because the source has a higher ratio than the background. Ultimately as the source becomes more depleted, there is a decline in the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio of the system.\u003c/p\u003e \u003cp\u003eConvex functions were created by reversing the sign of the \u003cem\u003ex\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e term in the original relationship and then negative and positive \u003cem\u003ey\u003c/em\u003e-intercepts were once again created by subtracting or adding 0.05 umoles/ml to the original \u003csup\u003e87\u003c/sup\u003eSr values. This also creates three functions with similar curvature differing only in their \u003cem\u003ey\u003c/em\u003e-intercepts. Note that a convex polynomial relationship defined by \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr plots suggests that the source is becoming more enriched with added material. Evaluation of these artificially created curvilinear datasets (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e2\u003c/span\u003eb) results in three different patterns than what are presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e2\u003c/span\u003ea: (A) if the \u003cem\u003ey\u003c/em\u003e-intercept is zero, the \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratio will increase as more Sr is added to the system, simply because the source is more enriched than the system itself. In this case, the original \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr curvature (based on a changing source) is the only driver for changes in the isotopic ratio. (B) If the \u003cem\u003ey\u003c/em\u003e-intercept is positive (indicating that the original source is less than the background), the ratio of \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr will initially decrease as more Sr is added to the system, but as the source becomes increasingly enriched, the ratio will begin to increase as the source becomes more influential than the background. (C) If the \u003cem\u003ey\u003c/em\u003e-intercept is negative, the ratio of \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr increases as more enriched Sr is added to the system. In this case, the increase in the ratio is most rapid initially where the ratio underestimates what function curvature suggests. Supplemental Material; Section 10 simulations of curvilinear denominator vs. numerator relationships also suggest that the non-zero \u003cem\u003ey\u003c/em\u003e-intercepts in polynomial functions are perfectly related to system backgrounds.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Quantifying Isotopic Sources\u003c/h2\u003e \u003cp\u003eAn introduction of isotopic sources is presented in Supplemental Material: Section 5, Quantifying isotopic sources and Supplemental Material and Section 6, More nuances of Keeling plots and mixing diagrams). Here we include source evaluations of material presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea-e.\u003c/p\u003e \u003cp\u003eA mixing diagram for the data presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea and c is presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e3\u003c/span\u003ea. For the 1/denominator (1/\u003csup\u003e86\u003c/sup\u003eSr in umole/ml units) form of the mixing diagrams, slopes match the \u003cem\u003ey\u003c/em\u003e-intercepts shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea and c. The \u003cem\u003ey\u003c/em\u003e-intercepts of the mixing diagrams match the slope of the original \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr plots shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea and c. As shown above, the \u003cem\u003ey\u003c/em\u003e-intercepts of \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr plots are related to backgrounds. Either a mixing diagram (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e3\u003c/span\u003ea) or an \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr plot can identify exogenous sources and relative backgrounds.\u003c/p\u003e \u003cp\u003eA different form of mixing diagram for the data in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea and c is presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e3\u003c/span\u003eb. For this example, the 1/total Sr concentration expression in umole/ml units is used as the \u003cem\u003ex\u003c/em\u003e-axis rather than the 1/\u003csup\u003e86\u003c/sup\u003eSr (in umole/ml units) used in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e3\u003c/span\u003ea. For the 1/total element (in umole/ml units) form of the mixing diagram, the slopes do not exactly match the \u003cem\u003ey\u003c/em\u003e-intercepts shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea and c. The mixing line slopes match denominator vs. numerator \u003cem\u003ey\u003c/em\u003e-intercepts if the 1/denominator form of the mixing line is used because the two equations are equivalent \u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;m\u003cem\u003ex\u003c/em\u003e\u0026thinsp;+\u0026thinsp;b and \u003cem\u003ey\u003c/em\u003e/\u003cem\u003ex\u003c/em\u003e = (1/\u003cem\u003ex\u003c/em\u003e) b\u0026thinsp;+\u0026thinsp;m. The \u003cem\u003ex\u003c/em\u003e terms differ for these two equations if the 1/total umoles element form (in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e3\u003c/span\u003eb) of the mixing line is used. The \u003cem\u003ey\u003c/em\u003e-intercepts (in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e3\u003c/span\u003ea and b) of the mixing diagrams that represents a source for both forms match the slopes of the original \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr plots shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea and c. This occurs because the denominator of an isotopic ratio and total element are almost perfectly related because the portion of total for the denominator varies over a miniscule range. The 1/total element values in either ug/ml or umoles/ml units are a surrogate for 1/\u003csup\u003e87\u003c/sup\u003eSr in umole/ml units. Conventional mixing diagrams in the isotope literature usually use 1/total Sr concentrations in mass units (ug/ml in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e3\u003c/span\u003ec) rather than molar units that are shown in both Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e3\u003c/span\u003ea and b). As stated above the \u003cem\u003ey\u003c/em\u003e-intercept of the 1/total element concentration in mass units mixing line form also correctly identifies the source. Only the 1/\u003csup\u003e86\u003c/sup\u003eSr form (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e3\u003c/span\u003ea) of mixing line (using molar values) produces slopes exactly matching \u003cem\u003ey\u003c/em\u003e-intercepts in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003ea and c.\u003c/p\u003e \u003cp\u003eDifferent background isotopic composition can cause mathematical-driven differences in \u003cem\u003ey\u003c/em\u003e-intercepts for denominator isotope vs. numerator isotope plots. However, just because \u003cem\u003ey\u003c/em\u003e-intercepts differ, does not necessarily suggest that background differences are the sole cause. The \u003cem\u003ey\u003c/em\u003e-intercepts of \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr regression equations are extremely small. Source detection, either from denominator isotope vs. numerator isotope slope; or indirectly from standard Keeling plots or mixing diagram \u003cem\u003ey\u003c/em\u003e-intercepts, are less prone to error and can be more accurately calculated than background isotopic composition.\u003c/p\u003e \u003cp\u003eLinear denominator isotope vs. numerator isotope functions (\u003cem\u003ey\u003c/em\u003e\u0026thinsp;=\u0026thinsp;m\u003cem\u003ex\u003c/em\u003e\u0026thinsp;+\u0026thinsp;b) must produce linear mixing diagrams (\u003cem\u003ey\u003c/em\u003e/\u003cem\u003ex\u003c/em\u003e = (1/\u003cem\u003ex\u003c/em\u003e) b\u0026thinsp;+\u0026thinsp;m). Detectable curvature in a Keeling plot or mixing diagram suggests that the original denominator isotope vs. numerator isotope function is not linear. Therefore, an evaluation for possible Keeling plot or mixing line curvature can supplement denominator vs. numerator evaluations for statistically significant \u003cem\u003ex\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e terms in polynomial best-fit equations, and spline smoothed cubic derivatives when assessing whether a denominator vs. numerator plot is linear.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4 Sources, Backgrounds and y-intercepts for Curvilinear Denominator vs. Numerator Functions\u003c/h2\u003e \u003cp\u003eThe purpose of the simulations (presented in Supplemental Material: Section 10, Simulations demonstrating source, background and \u003cem\u003ey\u003c/em\u003e-intercept effects on isotopic ratios for curvilinear denominator vs. numerator functions) presented here reiterate the fact that changing observed isotopic ratios are not necessarily related to changing sources. A second simulation goal is to demonstrate that interpreting a nonlinear mixing diagram, although difficult, can still be theoretically used for a changing source system, as data can be manipulated to deal with changing sources. A third goal was to demonstrate that even though \u003cem\u003ey\u003c/em\u003e-intercepts for curvilinear \u003csup\u003e86\u003c/sup\u003eSr vs. \u003csup\u003e87\u003c/sup\u003eSr relationships in real datasets may be difficult to evaluate, they are mathematically related to system backgrounds.\u003c/p\u003e \u003cp\u003e \u003cb\u003e2.5 Even if All Treatments in a Study are Defined with Linear Denominator vs. Numerator Functions, Treatments are not Necessarily Similarly Confounded.\u003c/b\u003e \u003c/p\u003e \u003cp\u003eThe size of the non-zero \u003cem\u003ey\u003c/em\u003e-intercept and the distance of data points from the origin affect isotopic ratios\u0026rsquo; confounding severity. The effects of different \u003cem\u003ey\u003c/em\u003e-intercepts and the position of points (both close to and farther away from the origin) can be clearly seen in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e1\u003c/span\u003eb and d. The distance from the origin is important because at large values of the denominator a ratio will approach the denominator vs. numerator slope. For treatments with similar slopes and positive denominator vs. numerator \u003cem\u003ey\u003c/em\u003e-intercepts, an average isotope ratio will be larger for a treatment with many low denominator values than for a treatment with higher denominator values.\u003c/p\u003e \u003cp\u003eThe Supplemental Material: Section 11, Simulation demonstrating why not all treatments in an experimental study are similarly confounded outlines how approximated quantitative numerical indicators of the degree of isotopic confounding can be theoretically defined. This is done by combining both \u003cem\u003ey\u003c/em\u003e-intercept and denominator size effects. A similar real data example from Moyo \u003cem\u003eet al.\u003c/em\u003e\u003csup\u003e25\u003c/sup\u003e reveals that \u003csup\u003e15\u003c/sup\u003eN:\u003csup\u003e14\u003c/sup\u003eN ratios are overestimated for all species in a trophic level study\u003csup\u003e16\u003c/sup\u003e Supplemental Material 2. While all six species have positive \u003cem\u003ey\u003c/em\u003e-intercepts for linear \u003csup\u003e14\u003c/sup\u003eN vs. \u003csup\u003e15\u003c/sup\u003eN plots, scaling-related overestimates are severe for only two of six species evaluated.\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Discussion","content":"\u003cp\u003eDifferences in isotopic ratio for various times, samples or treatments depend on three things for linear denominator vs. numerator relationships: First, the isotopic ratio of material being exogenously added to, or endogenously removed. Second, the initial isotopic ratio for the system to which material was exogenously added to, or endogenously removed from, and third, total element concentration, which is always strongly correlated to the denominator of an isotopic ratio. Factors (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e) and (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e) are respectively referred to as sources and backgrounds in isotopic studies\u003csup\u003e3,26\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eSince ratio values increase and approach the denominator vs. numerator slope for linear functions with negative \u003cem\u003ey\u003c/em\u003e-intercepts and decrease to approach the denominator vs. numerator slope for functions with positive \u003cem\u003ey\u003c/em\u003e-intercepts, a ratio will almost always underestimate (for negative \u003cem\u003ey\u003c/em\u003e-intercepts) or overestimate (for positive \u003cem\u003ey\u003c/em\u003e-intercepts) the denominator vs. numerator slope. These under and overestimates of the denominator vs. numerator slope will likely create ratios and slopes that are not strongly related.\u003c/p\u003e \u003cp\u003eKeeling plots and isotopic mixing diagrams are mathematically indirect ways to calculate the slope and relative \u003cem\u003ey\u003c/em\u003e-intercepts of linear denominator vs. numerator concentration isotope plots. Linear non-zero \u003cem\u003ey\u003c/em\u003e-intercepts are related to system backgrounds. Without knowing total element amounts, whether differing isotopic ratios suggest changing sources or simple dilution or concentration (as an observed ratio approaches the linear denominator vs. numerator slope as total element increases) cannot be determined. If the relationship between denominator isotope vs. numerator isotope is not linear, a fourth factor (denominator vs. numerator function curvature) also affects isotopic ratios, in ways (as shown above) that are intuitively difficult to predict.\u003c/p\u003e \u003cp\u003eSummaries of the interpretive consequences of non-zero \u003cem\u003ey\u003c/em\u003e-intercepts for both linear and curvilinear functions is presented in Supplemental Material: Section 12, Supplementary Table \u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003e and S2. Table \u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003e summarizes the mathematical consequences of likely ubiquitous non-zero, \u003cem\u003ey\u003c/em\u003e-intercepts for isotopic ratio denominator vs. isotopic ratio numerator relationships. The diagrams indicate the shape of the denominator vs. observed ratio plot, and the shape of the denominator isotopic concentration vs. numerator isotope concentration plot for nine function categories. For an easier reference, Table S2 presents the same graphics in Table \u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003e but without the narrative.\u003c/p\u003e \u003cp\u003eIsotopic discrimination can also lead to curvilinear denominator vs. numerator relationships. For example, if there is discrimination against a rarer heavier isotope, a substrate would become more enriched in the heavy isotope as it decomposes to a product. Either source changes or isotopic discrimination can create curvilinear relationships. Significant \u003cem\u003ex\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e terms in denominator vs. numerator polynomial best-fit equations, and consistently changing spline smoothed cubic derivatives are useful for curvature detection regardless of its cause.\u003c/p\u003e \u003cp\u003eMost importantly, changes in observed isotopic ratios do not necessarily reflect source changes. Furthermore, standard equations that convert isotopic ratios to the portion of total element that is derived from a source (such as N derived from fertilizer [NDFF]) do not account for the fact that isotopic ratios can be altered by total element concentration. Similarly, procedures to calculate discrimination factors\u003csup\u003e27\u003c/sup\u003e, if calculated from confounded isotopic ratios, are subject to error.\u003c/p\u003e \u003cp\u003eMany variables scale with body size\u003csup\u003e28\u003c/sup\u003e. Therefore, isotopic ratios could depend on total element content, and thus complicate standard isotopic interpretations\u003csup\u003e12\u003c/sup\u003e. For another example, it is possible that massive differences in tree tissue sizes (flower buds, flowers, leaves, branches, roots, structural wood) imply that total N content for different tissues may differ over several orders of magnitude. These size differences could affect \u003csup\u003e15\u003c/sup\u003eN:\u003csup\u003e14\u003c/sup\u003eN ratios and the NDFF values derived from these ratios. Without accessing total N contents it may be difficult to determine if different isotopic ratios represent real differences in physiological partitioning or are an indirect scaling effect.\u003c/p\u003e \u003cp\u003eWe emphasize that not all isotopic datasets are confounded. Patterns where a source changes with exogenous gains or endogenous losses can also be much more complex than the relatively simple constant source (linear) and consistently-changing source (polynomial) models illustrated here. However, one will never know if isotopic data is confounded unless it is evaluated. Isotopic researchers need encouragement to change. Funding agencies should not support isotopic research unless proposed studies include source analyses.\u003c/p\u003e"},{"header":"4. Methods","content":"\u003cp\u003eSince this first paper in a two-paper series deals with theoretical mathematical relationships that will be associated with misinterpretation in isotopic data, detailed methods are not included. All artificial data produced here have realistic data ranges and reflect what will be demonstrated in the companion publication\u003csup\u003e16\u003c/sup\u003e with real data. Sr isotopes were used to demonstrate mathematical principles because \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr ratios are less confusing for many statistically -oriented readers unfamiliar with the delta notation used for other common isotopes.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003ch2\u003eAdditional Informationl\u003c/h2\u003e \u003cp\u003eAdditional Information is included in Supplemental Material\u003c/p\u003e \u003c/p\u003e\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 the first draft of the manuscript once figure legends and tables were created by TLR, CPN and CN. 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\u003eAcknowledgements\u003c/h2\u003e \u003cp\u003eSincere thanks and love to TLR\u0026rsquo;s life Mary Ann Righetti (BS,MLS,JD). She has been a sounding board for the many years of concept development. Her non-STEM related evaluations improve clarity. Furthermore, she is the originator of the ideas and format behind Table\u0026nbsp;2, which goes a long way to making the manuscript comprehensible. Mary Ann also encouraged us to include the Martian data. 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 Statement\u003c/h2\u003e \u003cp\u003eUpon acceptance all data that is presented in Figures or Tables in both the main text or supplemental material will either be posted with a general-purpose archival service such as Dryad or included in supplementary material.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eBanner, J. L., Musgrove, M., Asmerom, Y., Edwards, R. L. \u0026amp; Hoff, J. A. High-resolution temporal record of Holocene ground-water chemistry: tracing links between climate and hydrology. Geology 24, 1049\u0026ndash;1053 (1996).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eEricson, J. E. Strontium isotope characterization in the study of prehistoric human ecology. J. Hum. Evol. 14, 503\u0026ndash;514 (1985).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eGrupe, G. \u003cem\u003eet al.\u003c/em\u003e Mobility of bell beaker people revealed by strontium isotope ratios of tooth and bone: a study of southern bavarian skeletal remains. Appl. Geochem. 12, 517\u0026ndash;525 (1997).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eGannes, L. Z., del Rio, C. M. \u0026amp; Koch, P. Natural abundance variations in stable isotopes and their potential uses in animal physiological ecology. Comp. Biochem. Physiol. A Mol. Integr. Physiol. 119, 725\u0026ndash;737 (1998).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eEhleringer, J. R., Hall, A. E. \u0026amp; Farquhar, G. D. \u003cem\u003eStable isotopes and plant carbon-water relations\u003c/em\u003e (Academic Press, 1993).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJasienski, M. \u0026amp; Bazzaz, F. A. The fallacy of ratios and the testability of models in biology. Oikos 84, 321\u0026ndash;326 (1999).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePhillips, D. L., and Koch, P. L. \u0026ldquo;Incorporating Concentration Dependence in Stable Isotope Mixing Models,\u0026rdquo; Ecologies, 130, 114\u0026ndash;125, (2002),\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePearson, S. F., Levey, D. J., and Greenberg, C. H., \u0026ldquo;Effects of Elemental Composition on the Incorporation of Dietary Nitrogen and Carbon Isotopic Signatures in an Omnivorous Songbird,\u0026rdquo; Oecologia, 135, 516\u0026ndash;523 (2003).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eRighetti, T. L., Dalthorp, D., Sandrock, D., Strik, B., Banados, P., and Zhou, Z. \u0026ldquo;Slope-Based and Ratio-Based Approaches to Determine Fertiliser-Derived N in Plant Tissues for Established Perennial Plants,\u0026rdquo; The Journal of Horticultural Science and Biotechnology, 82, 641\u0026ndash;647 (2007b).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eRighetti, T.L., Sandrock, D.R., Strik, B. and Azarenko, A. \u0026ldquo;Appropriate analysis and interpretation approaches to determine fertilizer-derived nitrogen in plant tissues.\u0026rdquo; Journal of the American Society for Horticultural Science, 132, 429\u0026ndash;436, (2007c\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eRighetti, T. L., Dalthorp D., Lambrinos, J. D., Strik, B., Sandrock, D., and Phillips, C. \u0026ldquo;Scaling Confounds the Interpretation of Isotopic Data,\u0026rdquo; International Journal of Environmental Analytical Chemistry, 92, 1\u0026ndash;27 (2012).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eTejada-Lara JV, MacFadden BJ, Bermudez L, Rojas G, Salas-Gismondi R, Flynn JJ. 2018 Body mass predicts isotope enrichment in herbivorous mammals. Proc. R. Soc. B 285: 2018\u0026thinsp;\u0026ndash;\u0026thinsp;1020 (2018).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eVillamar\u0026iacute;n, F., Jardine, T. D., Bunn, S. E., Marioni, B., and Magnusson, W. E. \u0026ldquo;Body Size is More Important Than Diet in Determining Stable-Isotope Estimates of Trophic Position in Crocodilians,\u0026rdquo; Scientific Reports, 8, 1\u0026ndash;11 (2018).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJacobi, C. M., Villamarin, F., Jardine, T. D., and Magnusson, W.E. \u0026ldquo;Uncertainities Associated with Trophic Discrimination Factor and Body Size Complicate Calculation of d\u003csup\u003e15\u003c/sup\u003eN-Derived Trophic Positions in Arapaima sp.,\u0026rdquo; Ecology of Freshwater Fish, 29, 779\u0026ndash;789 (2020).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003e, Whitledge, G. W., and Rabeni, C. F. \u0026ldquo;Energy Sources and Ecological Role of Crayfishes in an Ozark Stream: Insights from Stable Isotopes and Gut Analysis,\u0026rdquo; Canadian Journal of Fisheries and Aquatic Sciences, 54, 2555\u0026ndash;2563 (1997).\u003c/span\u003e\u003c/li\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: dealing with confounded isotopic ratios. Scientific Reports \u003cem\u003ethis issue\u003c/em\u003e (2024).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKratochv\u0026iacute;l, L. \u0026amp; Flegr, J. Differences in the 2nd to 4th digit length ratio in humans reflect shifts along the common allometric line. Biol. Lett. 5, 643\u0026ndash;646 (2009).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePackard, G. C. \u0026amp; Boardman, T. J. The misuse of ratios, indices, and percentages in ecophysiological research. Physiol. Zool. 61, 1\u0026ndash;9 (1988).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eRighetti, T. L. \u003cem\u003eet al.\u003c/em\u003e Analysis of ratio-based responses. J. Am. Soc. Hortic. Sci. 132, 3\u0026ndash;13 (2007).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eTanner, J. M. Fallacy of per-weight and per-surface area standards, and their relation to spurious correlation. J. Appl. Physiol. 2, 1\u0026ndash;15 (1949).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSagan, C. Encyclopaedia galactica in \u003cem\u003eCosmos: a personal voyage. Episode 12\u003c/em\u003e 24 (PBS, 1980).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKeeling, C. D. The concentration and isotopic abundances of atmospheric carbon dioxide in rural areas. Geochim. Cosmochim. Acta 13, 322\u0026ndash;334 (1958).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePataki, D. E. \u003cem\u003eet al.\u003c/em\u003e The application and interpretation of Keeling plots in terrestrial carbon cycle research. Glob. Biogeochem. Cycles 17, 1022 (2003).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eGoede, A., McCulloch, M., McDermott, F. \u0026amp; Hawkesworth, C. Aeolian contribution to strontium and strontium isotope variations in a Tasmanian speleothem. Holocene 9, 715\u0026ndash;722 (1998).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMoyo, S. \u003cem\u003eet al.\u003c/em\u003e Stable isotope analyses identify trophic niche partitioning between sympatric terrestrial vertebrates in coastal saltmarshes with differing oiling histories. PeerJ 9, e11392 (2021).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBilby, R. E., Fransen, B. R. \u0026amp; Bisson, P. A. Incorporation of nitrogen and carbon from spawning coho salmon into the trophic system of small streams: evidence from stable isotopes. Can. J. Fish. Aquat. Sci. 53, 164\u0026ndash;173 (1996).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMariotti, A. \u003cem\u003eet al.\u003c/em\u003e Experimental determination of nitrogen kinetic isotope fractionation: some principles; illustration for the denitrification and nitrification processes. Plant Soil 62, 413\u0026ndash;430 (1981).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSavage, V. M. \u003cem\u003eet al.\u003c/em\u003e The predominance of quarter-power scaling in biology. Funct. Ecol. 18, 257\u0026ndash;282 (2004).\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-4086468/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4086468/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eStable mass isotopic ratios (such as \u003csup\u003e13\u003c/sup\u003eC:\u003csup\u003e12\u003c/sup\u003eC, \u003csup\u003e15\u003c/sup\u003eN:\u003csup\u003e14\u003c/sup\u003eN, \u003csup\u003e18\u003c/sup\u003eO:\u003csup\u003e16\u003c/sup\u003eO \u003csup\u003e87\u003c/sup\u003eSr:\u003csup\u003e86\u003c/sup\u003eSr and \u003csup\u003e34\u003c/sup\u003eS:\u003csup\u003e32\u003c/sup\u003eS) are used to interpret archaeological, climate change, ecological, geological, and physiological studies. Most isotopic reports evaluate changes in observed isotopic ratios or ratio-based expressions over time or among treatments. To address concerns that ratios or ratio-based expressions may not produce conclusions that support known physiological or ecological principles, source (isotopic ratio of the material being added or lost) analyses are proposed as an alternative to statistically analysing observed isotopic ratios. Mathematically defined relationships between observed ratios, backgrounds (isotopic ratio of a system before any loses or gains), sources and total element concentrations as well as denominator vs. numerator relationships are presented. These mathematical relationships suggest that source-based approaches often produce conclusions that differ from ratio-based evaluations. Total element concentrations, necessary for source analyses, are presented in less than half of isotopic publications. Without evaluating total element, relative background and source ratios cannot be determined. Even, when total element data is available, researchers rarely conduct source analyses. This is unfortunate because determining sources solves most interpretive issues. Our goal is to advocate better methods when analyzing isotopic ratios in the thousands of mass isotope publications annually produced.\u003c/p\u003e","manuscriptTitle":"A Mathematical Explanation for Why Ratio-Based Isotopic Analyses are Commonly Misleading: Theory","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-03-26 06:27:42","doi":"10.21203/rs.3.rs-4086468/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":"March 26th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":29838035,"name":"Biological sciences/Biological techniques"},{"id":29838036,"name":"Earth and environmental sciences/Biogeochemistry"},{"id":29838037,"name":"Earth and environmental sciences/Climate sciences"},{"id":29838038,"name":"Earth and environmental sciences/Ecology"},{"id":29838039,"name":"Earth and environmental sciences/Environmental sciences"},{"id":29838040,"name":"Physical sciences/Chemistry"}],"tags":[],"updatedAt":"2024-05-22T04:29:23+00:00","versionOfRecord":[],"versionCreatedAt":"2024-03-26 06:27:42","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4086468","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4086468","identity":"rs-4086468","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}