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\begin{document}

\title{Experimental assessment of EAIRMS normalization methodologies for
environmental stable isotopes}



\author[1]{Sawyer Balint}%
\author[1]{Morgan Schwartz}%
\author[2]{Stella Linnekogel}%
\author[3]{Drew Fowler}%
\author[4]{Sáde Cromratie Clemons}%
\author[5]{Laura K. Burkemper}%
\affil[1]{US Environmental Protection Agency Office of Research and Development Atlantic Coastal Environmental Sciences Division}%
\affil[2]{Centre for Ecology \& Hydrology Lancaster Site}%
\affil[3]{US Geological Survey Lower Mississippi-Gulf Water Science Center Baton Rouge LA Office}%
\affil[4]{University of Colorado Boulder Department of Geography}%
\affil[5]{The University of New Mexico}%


\vspace{-1em}



  
  \date{May 25, 2024}


\begingroup
\let\center\flushleft
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\selectlanguage{english}
\begin{abstract}
\textbf{RATIONALE}: In stable isotope mass spectrometry, isotope values
are normalized to internationally recognized reference scales using
certified reference materials and working standards. Numerous techniques
exist for performing this normalization, but these methodologies need to
be experimentally assessed to compare their impact on reproducibility of
isotope results. \textbf{METHODS}: We tested normalization methods by
the number of standards used, their matrix, their isotope range, and
whether normalization required extrapolating beyond the isotope range.
Using 8 certified reference materials and 5 working standards on a
ThermoFinnigan Delta-V IRMS and Elementar VisION IRMS for nitrogen and
carbon isotope composition via solid combustion with an elemental
analyzer, we computed every possible isotope normalization (n=6272).
Additionally, we assessed how sample matrix impacted linearity effects
on both instruments. \textbf{RESULTS}: Normalizations composed of three
or four reference materials had better performance than one-point and
two-point methods, especially when the normalization was matrix-mixed or
extrapolated, and normalizations with an isotope range greater than 15%0
were more accurate under these conditions. Normalizations that were
matrix-matched and were not extrapolated exhibited the highest accuracy.
Linearity effects were found to exceed instrument precision by two
orders of magnitude irrespective of sample matrix and were not predicted
by reference gas diagnostics. \textbf{CONCLUSIONS}: To maximize
interlaboratory comparability of isotope results, operators of EAIRMS
systems should use at least 3 calibration standards to construct their
normalizations, select standards with a large isotope range to avoid
extrapolation, and match the matrix of their standards to their samples
to the best extent possible.%
\end{abstract}%



\sloppy


\textbf{Experimental assessment of EAIRMS normalization methodologies
for environmental stable isotopes}

Sawyer Balint\textsuperscript{1}*, Morgan Schwartz\textsuperscript{2}*,
Drew Fowler\textsuperscript{3+}, Stella Linnekogel\textsuperscript{4+},
S\selectlanguage{ngerman}áde Cromratie Clemons\textsuperscript{5}, Laura K.
Burkemper\textsuperscript{6}

\textsuperscript{1}ORISE Participant, U.S. EPA Atlantic Coastal
Environmental Sciences Division, Narragansett, RI. 02882

\textsuperscript{2}U.S. EPA Atlantic Coastal Environmental Sciences
Division, Narragansett, RI. 02882

\textsuperscript{3}U.S. Geological Survey, Louisiana Cooperative Fish
and Wildlife Research Unit, Baton Rouge, LA. 70810

\textsuperscript{4}UK Centre for Ecology \& Hydrology Lancaster, Library
Avenue, Bailrigg, Lancaster, LA1 4AP, United Kingdom

\textsuperscript{5}Department of Geography, University of Boulder,
Boulder, CO

\textsuperscript{6}Center for Stable Isotopes, University of New Mexico,
Albuquerque, NM

* Authors share first authorship

\selectlanguage{english}\textsuperscript{+}Authors contributed equally to this manuscript

\textbf{Corresponding Author} :

Sawyer Balint

sjbalint@bu.edu

(518) 258-9510

Morgan Schwartz schwartz.morgan@epa.gov

(401) 782-9624

\textbf{CRediT author statement} :

\textbf{Sawyer Balint} : Conceptualization, Methodology, Software,
Validation, Formal analysis, Investigation, Data Curation, Writing --
Original Draft, Visualization, Supervision, Project administration.

\textbf{Morgan Schwartz} : Conceptualization, Methodology, Validation,
Formal analysis, Investigation, Resources, Data Curation, Writing --
Original Draft, Supervision, Project administration.

\textbf{Drew Fowler} : Conceptualization, Methodology, Investigation,
Writing -- Review \& Editing.

\textbf{Stella Linnekogel} : Conceptualization, Methodology,
Investigation, Writing -- Review \& Editing.

\textbf{Sade Cromratie Clemons:} Conceptualization, Methodology,
Investigation, Writing -- Review \& Editing.

\textbf{Laura K. Burkemper:} Conceptualization, Methodology, Validation,
Investigation, Resources, Data Curation, Writing -- Review \& Editing,
Supervision, Project administration.

\subsection*{Acknowledgements:}

{\label{acknowledgements}}

The authors would like to thank Viorel Atudorei, Seth Newsome, and
Zachary Sharp for their assistance in the planning and execution of
IsoCamp 2022, where the experimental design for this project was
conceived. Additionally, the authors thank Renee Brooks, Michaela
Cashman, and Autumn Oczkowski for review and feedback on the manuscript.
Stella Linnekogel (UKCEH) acknowledges support from the NERC project
Diurnal Variation in Soil Nitrous oxide Emissions (DIVINE): drivers and
mechanisms (NE/V000837/1). This article has been subjected to Agency
review and has been approved for publication. This work was supported in
part by an appointment to the Research Participation Program at the
Office for Research and Development, Center for Environmental
Measurement and Modelling, Atlantic Coastal Environmental Sciences
Division, U.S. Environmental Protection Agency (U.S. EPA), administered
by the Oak Ridge Institute for Science and Education through an
interagency agreement between the U.S. Department of Energy and U.S.
EPA. The views expressed in this presentation are those of the authors
and do not necessarily represent the views or polices of the U.S. EPA.
This article has been peer reviewed and approved for publication
consistent with USGS Fundamental Science Practices
(https://pubs.usgs.gov/circ/1367/). Any use of trade, firm, or product
names is for descriptive purposes only and does not imply endorsement by
the U.S. Government.

\subsection*{Abstract}

{\label{abstract}}

\textbf{RATIONALE} : In stable isotope mass spectrometry, isotope values
are normalized to internationally recognized reference scales using
certified reference materials and working standards. Numerous techniques
exist for performing this normalization, but these methodologies need to
be experimentally assessed to compare their impact on reproducibility of
isotope results.

\textbf{METHODS} : We tested normalization methods by the number of
standards used, their matrix, their isotope range, and whether
normalization required extrapolating beyond the isotope range. Using 8
certified reference materials and 5 working standards on a
ThermoFinnigan Delta-V IRMS and Elementar VisION IRMS for nitrogen and
carbon isotope composition via solid combustion with an elemental
analyzer, we computed every possible isotope normalization (n=6272).
Additionally, we assessed how sample matrix impacted linearity effects
on both instruments.

\textbf{RESULTS} : Normalizations composed of three or four reference
materials had better performance than one-point and two-point methods,
especially when the normalization was matrix-mixed or extrapolated, and
normalizations with an isotope range greater than 15\selectlanguage{english}%0 were more accurate
under these conditions. Normalizations that were matrix-matched and were
not extrapolated exhibited the highest accuracy. Linearity effects were
found to exceed instrument precision by two orders of magnitude
irrespective of sample matrix and were not predicted by reference gas
diagnostics.

\textbf{CONCLUSIONS} : To maximize interlaboratory comparability of
isotope results, operators of EAIRMS systems should use at least 3
calibration standards to construct their normalizations, select
standards with a large isotope range to avoid extrapolation, and match
the matrix of their standards to their samples to the best extent
possible.

\subsection*{}

{\label{section}}

Stable isotope mass spectrometry is a growing tool across disciplines,
including in biology where the nitrogen (N) and carbon (C) isotope
composition of solid samples are routinely used for investigating a
variety of physiological, ecological, and biogeochemical
questions\textsuperscript{1}. In recent decades, the broad applications
of stable isotopes combined with faster and more accessible
instrumentation have led to a large growth in the utilization and
publication of stable isotope measurements as they relate to the natural
sciences\textsuperscript{2}. Despite the growing importance of this
tool, immense variations in analytical methodology exist within the
field, leading to poor interlaboratory
comparability\textsuperscript{3}and complicating the interpretation and
reproducibility of scientific studies that use stable
isotopes\textsuperscript{4}.

Continuous flow stable isotope mass spectrometry\textsuperscript{5} via
solid combustion\textsuperscript{6,7} is the predominant analytical
technique used to quantify the C and N isotope composition of solid
samples for biological applications\textsuperscript{2}. Samples are
delivered to the isotope ratio mass spectrometer (IRMS) in a gaseous
form through a peripheral elemental analyzer (EA) and the isotope
composition is determined relative to a working gas that is injected
sequentially with the sample gas\textsuperscript{8,9}. This methodology
allows for high precision measurements, but to facilitate
interlaboratory comparison the results must be normalized to
internationally referenced isotope scales\textsuperscript{10-12}.
Contemporary normalization methods call for standard reference materials
to be processed identically to the unknown samples using the ``identical
treatment'' principle\textsuperscript{5,13}, and analyzed in tandem with
unknown samples\textsuperscript{14-16}, a normalization curve is then
computed with a least-squares linear regression\textsuperscript{17}. As
the number of available certified standards has increased in recent
decades\textsuperscript{18-20}, operators now have a multitude of
certified standards to choose from when performing their isotope
normalizations in addition to their own working standards.

Unfortunately, the reproducibility of normalization techniques has not
advanced at the same pace as the expanding use of stable isotopes.
Although modeling work has suggested that normalization error generally
decreases with the number of standards\textsuperscript{21}, stable
isotope laboratories vary immensely in the number of standards used due
in part to limited experimental assessments of related normalization
error. Furthermore, users of elemental analyzer isotope ratio mass
spectrometry (EAIRMS) for biological applications often analyze sample
matrixes for which certified reference materials do not exist (e.g.,
sediment\textsuperscript{22}, marine algae\textsuperscript{23}, samples
imbedded on glass fiber filters\textsuperscript{24}), but the impact of
mixing organic matrixes between samples and standards on normalization
accuracy is unknown. Whether standards should be selected to maximize
isotope range or closely bracket the unknown samples is another decision
that, absent of an experimental assessment across a variety of
normalization methods, is left to anecdotal procedures that may vary
between laboratories. Some analyses, such as those that incorporate N
tracer as part of the methodology\textsuperscript{25}, may require
extrapolation beyond the range of the normalization curve -- again with
unknown consequences. Ultimately, readers of studies that incorporate
stable isotopes are left to determine the reproducibility of the study
for themselves -- assuming the particulars of the normalization are even
included in the methods. This matter becomes increasingly more difficult
in the era of big data and meta-analysis, as multiple studies using
multiple methods are integrated and consequently compared directly.

Here, we aim to better quantify best normalizations practices for EAIRMS
analysis using an experimental assessment of a variety of normalization
methods using certified reference materials analyzed on two instrument
systems in two laboratories. We specifically assess how the accuracy of
the normalization is impacted by the number of standards, their matrix
relative to the unknown samples, their isotope range, and whether the
normalization requires extrapolation beyond the isotope range. We
further investigate how instrument linearity effects are impacted by
sample matrix through the first known interlaboratory comparison of
EAIRMS linearity effects. By developing a refined understanding of
EAIRMS best practices, we hope to better facilitate the reproducibility
of biological stable isotope applications.

\subsection*{Number of standards used to construct the
normalization}

{\label{number-of-standards-used-to-construct-the-normalization}}

Although a variety of methodologies have been used to normalize results
from isotope ratio mass spectroscopy\textsuperscript{8,15}, this work
focuses on the most common methods used in continuous flow applications:
one-point anchoring, two-point linear normalization, or multipoint
linear normalization \textsuperscript{2,17}. Typically, the IRMS
software will perform a one-point anchoring relative to the working gas
according to the following equations:\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\delta_{\text{raw}\left(\text{sample}\right)}=\frac{R_{\text{measured}(\text{sample})}}{R_{\text{measured}(\text{wg})}}-1\) & Eq. 1\tabularnewline
\midrule
\endhead
\(\delta_{\text{true}\left(\text{sample}\right)}=\delta_{\text{raw}\left(\text{sample}\right)}+\delta_{\text{true}\left(\text{wg}\right)}+\left(\delta_{\text{raw}\left(\text{sample}\right)}\times\delta_{\text{true}\left(\text{wg}\right)}\right)\) & Eq. 2\tabularnewline
\bottomrule
\end{longtable}

where \(\delta_{\text{raw}\left(\text{sample}\right)}\) is the measured isotope composition
(\emph{R\textsubscript{measured}} ) of the sample relative to the
working gas (\emph{wg} ) and\(\delta_{\text{true}\left(\text{wg}\right)}\) is the actual isotope
composition of the working gas specified by the operator. Per
convention, the relative difference of isotope ratios (\selectlanguage{greek}\emph{δ} \selectlanguage{english}) for N
and C are expressed in parts per thousand (%0)\textsuperscript{26}.
Although it is possible to quantify\(\delta_{\text{true}\left(\text{wg}\right)}\) and thus calculate
the true isotope composition of the sample\((\delta_{\text{true}\left(\text{sample}\right)})\) via a
one-point anchoring using the working gas alone, identical treatment
principle dictates replacing Eq. 2 with replicate analyses of combusted
reference material according to the following equation:\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\delta_{\text{true}\left(\text{sample}\right)}=\delta_{\text{raw}\left(\text{sample}\right)}+\left(\frac{\sum\delta_{\text{true}\left(\text{std}\right)}-\sum\delta_{\text{raw}\left(\text{std}\right)}}{n}\right)\) & Eq. 3\tabularnewline
\bottomrule
\end{longtable}

where \(\delta_{\text{raw}\left(\text{std}\right)}\) is the isotope composition of the standard
relative to the reference gas (Eq. 1) and\(\delta_{\text{true}\left(\text{std}\right)}\) is the
certified isotope composition of the standard. When two or more
standards are used for normalization as recommended by the International
Atomic Energy Agency (IAEA)\textsuperscript{11,27},\(\delta_{\text{true}\left(\text{sample}\right)}\)
can be calculated using a least squares regression
between\(\delta_{\text{true}\left(\text{std}\right)}\) and\(\delta_{\text{raw}\left(\text{std}\right)}\) according to the
following equations:\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(m=\left(\frac{n\sum{{(\delta}_{\text{raw}\left(\text{std}\right)}\times\delta_{\text{true}\left(\text{std}\right)})-\sum\delta_{\text{raw}\left(\text{std}\right)}\sum\delta_{\text{true}\left(\text{std}\right)}}}{n\sum\left(\delta_{\text{raw}\left(\text{std}\right)}\right)^{2}-\left(\sum\delta_{\text{raw}\left(\text{std}\right)}\right)^{2}}\right)\) & Eq. 4\tabularnewline
\midrule
\endhead
\(b=\left(\frac{\sum\delta_{\text{true}\left(\text{std}\right)}-m\sum\delta_{\text{raw}\left(\text{std}\right)}}{n}\right)\) & Eq. 5\tabularnewline
\(\delta_{\text{true}\left(\text{sample}\right)}=\delta_{\text{raw}\left(\text{sample}\right)}\times m+b\) & Eq. 6\tabularnewline
\bottomrule
\end{longtable}

where \emph{m} is the expansion coefficient, \emph{b} is the intercept,
and \emph{n} is the number of measurements of standards. Past work
investigating C isotopes has shown that normalizations using two or more
standards substantially reduce normalization errors compared to
one-point anchoring\textsuperscript{11,17} because the expansion
coefficient is rarely equal to one when multipoint normalizations are
used\textsuperscript{21}, but Eq. 3 assumes an expansion coefficient of
one. The addition of supplementary standards to produce a three- or
four-point normalization allows a coefficient of determination
(r\textsuperscript{2}) to be calculated and theoretically reduces the
extrapolation of errors stemming from isotope heterogeneity of the
standards, although experimental assessments\textsuperscript{17} and
Monte Carlo simulations\textsuperscript{21} of this hypothesis have so
far been limited to C. Furthermore, past assessments of how many
standards should be used in a normalization have not concurrently
considered the effects of scenarios such as matrix mixing and
extrapolation, which may require additional standards to ensure
acceptable results.

Finally, we note that single or multipoint anchoring can also be
accomplished by analyzing one or multiple isotope standards to calculate
an intercept (Eq. 5) and applying an expansion coefficient (\emph{m} )
calculated from a previous analysis of multiple standards to conduct a
pseudo-linear normalization (Eq. 6) assuming that the expansion
coefficient is stable over time\textsuperscript{17}. This normalization
method is susceptible to changes in instrument performance over days or
weeks and with different instrument tunings and is thus not investigated
in this work. Here, we perform one-point anchoring and two-point,
three-point, and four-point linear normalizations for N and C on two
EAIRMS systems in two laboratories to experimentally assess their
accuracy and interlaboratory comparability.

\subsection*{1.2 Standard matrix, normalization isotope range, and
instrument
linearity}

{\label{standard-matrix-normalization-isotope-range-and-instrument-linearity}}

The impact of sample matrix and isotope range of the standards on
normalization accuracy have not been as thoroughly investigated as the
number of standards. Past work has identified matrix effects between
organic and inorganic standards for N\textsuperscript{28}, but
laboratories that process biological samples frequently analyze a
variety of organic sample matrixes (e.g., muscle
tissue\textsuperscript{29}, plants\textsuperscript{30},
soils\textsuperscript{22}) that have different preparation techniques
and combustion properties. Although proper application of the
``identical treatment'' protocol would suggest matching the matrix of
the sample and of the standard\textsuperscript{31}, some sample
matrixes, such as sediment and particulate organic matter collected on
glass fiber filters, do not have corresponding certified reference
materials available for purchase, which thwarts matrix matching. The
effect of matrix on the isotope results of organic samples complicates
the interpretation of biological stable
isotopes\textsuperscript{32,33}and requires additional experimental
assessment.

The isotope range of the normalization curve, and how it may affect the
accuracy of results, presents another area of uncertainty. Laboratories
may use standards that have very different isotope compositions, thus
generating a normalization curve with a large isotope range (i.e.,
\textgreater{}40%0). Conversely, analysts may tightly bracket their
samples by using normalization curves with a comparatively small isotope
range. Furthermore, when a sample has an isotope composition that
exceeds the range of normalization standards, extrapolation beyond the
normalization curve may be required. Past work on C isotopes has
suggested that tightly bracketing unknown samples may marginally improve
accuracy if the normalization is not extrapolated\textsuperscript{21},
but this investigation was constrained to two-point C normalizations. In
this study, we explore how the matrix and isotope range of standards
impacts the accuracy of one-point, two-point, three-point, and
four-point normalizations for N and C.

The impact of instrument linearity and subsequent linearity corrections
is a final aspect of stable isotope analyses that may hinder the
reproducibility of biological studies. Linearity, the mass-dependent
change in reported isotope composition from an IRMS, is anecdotally well
known and yet has not been subjected to an experimental interlaboratory
comparison\textsuperscript{34}. This characteristic of IRMS systems is
of particular importance for biological applications because there can
be large variations in the organic matter content of the samples,
particularly for sediments and particulate organic matter. Because the
amount of N\textsubscript{2} and CO\textsubscript{2} that enters the
IRMS is proportional to the amount of organic matter in biological
samples, instrument linearity can become the most important determinant
of analysis precision for samples with high variations in organic matter
content. Operators of EAIRMS systems frequently develop their own
linearity corrections, if they correct for the phenomenon at all, but
the magnitude and variability of instrument linearity between facilities
is unknown. Whether linearity is dependent on the matrix being analyzed
or can be predicted from reference gas diagnostics are additional areas
of uncertainty. We seek to better understand how linearity affects
EAIRMS isotope results for biological applications by performing
reference gas diagnostics and replicate analyses of working standards of
different organic matter compositions and sample weight at two
laboratories.

\subsection*{Methods}

{\label{methods}}

\begin{enumerate}
\item ~
  \subsection*{Stable isotope analysis}

  {\label{stable-isotope-analysis}}
\end{enumerate}

Eight certified reference materials and five in-house working standards
were analyzed at the University of New Mexico Center for Stable Isotopes
(UNM-CSI) and the United States Environmental Protection Agency,
Atlantic Coastal Environmental Sciences Division (U.S. EPA). These
standards encompassed a \selectlanguage{greek}\emph{δ} \selectlanguage{english}\textsuperscript{15}N range of -4.52%0
to +37.83%0 and a \selectlanguage{greek}\emph{δ} \selectlanguage{english}\textsuperscript{13}C range of -35.05%0 to
-1.17%0 (Fig. 1), and the sample matrix and reference information is
included in Table 1. For the normalization analysis, the eight certified
reference materials were weighed to between 0.15 and 3.3 mg depending on
the N content of the sample matrix to provide \textasciitilde{}0.05 mg
of N. Dilutions were then computed for each matrix to provide
\textasciitilde{}0.04 mg C to the IRMS. Each standard was run 4 times at
each facility and were run in an alternating order to mitigate the
effects of instrument drift. To quantify instrument linearity, the five
working standards were weighed to between 0.09 and 40 mg and analyzed
using a constant matrix-specific C dilution to produce a range of
0.013-0.16 mg of N and 0.009-0.10 mg of C across all working standards.
At both facilities, the elemental analyzer was configured to inject a
longer O\textsubscript{2} dosing for standards with plant matrixes
(USGS90, USGS91, CSI Blue Grama, and CSI Green Chile) and soil matrixes
(CSI Soil). Analyses at UNM-CSI were performed on a Costech 4010
combustion EA paired with a ThermoFinnigan Delta-V continuous-flow IRMS
(by a Conflo IV), while analyses at the U.S. EPA were performed on an
Elementar Vario Isotope Select EA paired with an Elementar VisION
continuous-flow IRMS. The pooled standard deviation (\selectlanguage{ngerman}±1\selectlanguage{greek}σ\selectlanguage{english}) of the
certified reference materials were \selectlanguage{ngerman}± 0.115 for\selectlanguage{greek}\emph{δ}
\selectlanguage{english}\textsuperscript{15}N and \selectlanguage{ngerman}±0.044 for\selectlanguage{greek}\emph{δ} \selectlanguage{english}\textsuperscript{13}C at
the U.S. EPA, and \selectlanguage{ngerman}± 0.084 for\selectlanguage{greek}\emph{δ} \selectlanguage{english}\textsuperscript{15}N and \selectlanguage{ngerman}±0.069
for\selectlanguage{greek}\emph{δ} \selectlanguage{english}\textsuperscript{13}C at UNM-CSI. Pooled standard deviations
of each certified reference standard are shown in Table 1.

\textbf{Table 1} : Certified isotope composition and associated
uncertainty (\selectlanguage{ngerman}±1\selectlanguage{greek}σ\selectlanguage{english}) of certified standard reference materials used in this
study.\selectlanguage{english}
\begin{longtable}[]{@{}lllllll@{}}
\toprule
\begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
Name\strut
\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
Certified \selectlanguage{greek}\emph{δ}\selectlanguage{english}\textsuperscript{15}N (%0 \selectlanguage{ngerman}±1\selectlanguage{greek}σ\selectlanguage{english})\strut
\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
Certified \selectlanguage{greek}\emph{δ}\selectlanguage{english}\textsuperscript{13}C (%0\selectlanguage{ngerman}±1\selectlanguage{greek}σ\selectlanguage{english})\strut
\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
Observed \selectlanguage{greek}\emph{δ}\selectlanguage{english}\textsuperscript{15}N (\selectlanguage{ngerman}±1\selectlanguage{greek}σ\selectlanguage{english})\strut
\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
Observed \selectlanguage{greek}\emph{δ}\selectlanguage{english}\textsuperscript{13}C (\selectlanguage{ngerman}±1\selectlanguage{greek}σ\selectlanguage{english})\strut
\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
Matrix\strut
\end{minipage} & \begin{minipage}[b]{0.14\columnwidth}\raggedright\strut
Matrix Classification\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
IAEA600\textsuperscript{11}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
+1.0 \selectlanguage{ngerman}± 0.2\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
--27.771 \selectlanguage{ngerman}± 0.043\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.063\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.046\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Caffeine\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
High Organic\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
USGS40\textsuperscript{18}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
--4.52 \selectlanguage{ngerman}± 0.06\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
--26.39 \selectlanguage{ngerman}± 0.04\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.041\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.067\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
L-glutamic acid\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
High Organic\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
USGS61\textsuperscript{19}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
--2.87 \selectlanguage{ngerman}± 0.04\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
--35.05 \selectlanguage{ngerman}± 0.04\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.027\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.042\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Caffeine\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
High Organic\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
USGS63\textsuperscript{19}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
+37.83 \selectlanguage{ngerman}± 0.06\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
--1.17 \selectlanguage{ngerman}± 0.04\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.052\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.120\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Caffeine\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
High Organic\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
USGS88\textsuperscript{20}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
+14.96 \selectlanguage{ngerman}± 0.14\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
--16.06 \selectlanguage{ngerman}± 0.07\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.046\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.092\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Collagen\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
High Organic\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
USGS89\textsuperscript{20}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
+6.25 \selectlanguage{ngerman}± 0.12\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
--18.13 \selectlanguage{ngerman}± 0.11\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.142\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.094\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Collagen\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
High Organic\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
USGS90\textsuperscript{20}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
+8.84 \selectlanguage{ngerman}± 0.17\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
--13.75 \selectlanguage{ngerman}± 0.06\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.052\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.210\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Flour\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Plant\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
USGS91\textsuperscript{20}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
+1.78 \selectlanguage{ngerman}± 0.12\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
-- 28.28 \selectlanguage{ngerman}± 0.08\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.048\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\selectlanguage{ngerman}±0.111\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Flour\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Plant\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
CSI Blue Grama\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Plant tissue\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Plant\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
CSI Casein\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Protein\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
High Organic\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
CSI Tuna\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Muscle tissue\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
High Organic\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
CSI Chile\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Plant tissue\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Plant\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
CSI Soil\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Sediment\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Sediment\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

\textbf{Fig. 1} : Isotope range of certified reference materials and
working standards used in this study\textsuperscript{11,18-20}

\subsection*{2.2 Statistical analyses}

{\label{statistical-analyses}}

To test normalization accuracy, we only used the 8 certified reference
materials as calibration standards, while the laboratory working
standards were used to test linearity in the two instruments. The
isotope composition of the certified and working standards were
normalized to the working gas (Eq. 1) in the vendor IRMS software
(Isodat and lyticOS for the Thermo Delta-V and Elementar VisION,
respectively) and then exported in a tabular format; all subsequent
normalizations and analyses (Eq. 3-5) were performed in R version
4.2.1\textsuperscript{35}. For each normalization, two certified
reference materials were designated as quality controls. Quality
controls were excluded from the normalization calculation, and the
performance of the normalization was assessed using the average observed
isotope composition of the quality controls relative to their expected
value\textsuperscript{36}. For each combination of quality controls (28
unique combinations), all possible one-point, two-point, three-point,
and four-point combinations of the remaining certified reference
materials were determined for a total of 1568 combinations. These
remaining certified reference materials were used as calibration
standards. Using those combinations of calibration standards and quality
controls, one-point anchoring and multipoint linear normalizations were
calculated for each element (C and N) and facility (2) for a total of
6272 normalizations. Two-point normalizations composed of IAEA 600 and
USGS 91 were excluded from subsequent data analysis and visualization
because the small isotope range between those standards (\textless{}1%0)
precluded an accurate calculation of a realistic expansion coefficient.
Although these standards could be used for a two-point anchoring using
an expansion coefficient derived from a different multipoint
normalization, assessing that method is beyond the scope of this study.

To assess how instrument accuracy was impacted by the selection of
standards and quality controls, the normalizations were characterized
according to their isotope range, the matrix of the standards relative
to the quality controls, and whether the normalization was extrapolated.
The isotope range of each normalization was calculated for each element
as the difference between the maximum and minimum expected isotope
composition of the calibration standards used in that normalization. If
the expected isotope value of both quality controls fell outside the
isotope range of the calibration standards, then the normalization was
classified as an ``extrapolation''. One-point normalizations, which have
an isotope range of zero, were classified as an ``extrapolation'' if the
single calibration standard was not bracketed by the two quality
controls. Finally, the matrix of each standard was classified as high
organic (i.e., protein, caffeine, collagen, L-glutamic acid) or plant
(i.e., plant tissue, flour). If the matrix of the calibration standards
matched the matrix of the quality controls than the normalization was
classified as ``matrix matched'', while if the matrix of the calibration
standards and quality controls were different (e.g., high organic
standards used to normalize plant quality controls), then the
normalization was classified as ``matrix mixed''. If both the quality
controls and the calibration standards were composed of a combination of
plants and high organics, then the normalization was classified as
``both mixed.''

The significance of differences between different normalization
methodologies, facilities, matrixes, and extrapolation status were
assessed using Kruskal-Wallis testing with Dunns post-hoc testing after
the assumption of normality was rejected with Shapiro-Wilks's
testing\textsuperscript{35}.

\subsection*{Results}

{\label{results}}

\begin{enumerate}
\item ~
  \subsection*{Normalization methodology
  comparison}

  {\label{normalization-methodology-comparison}}
\end{enumerate}

The impact of the number of calibration standards on normalization
errors were considered for two subsets of the data: normalizations that
were matrix-matched and bounded, and normalizations that were
matrix-mixed and extrapolated. The former set of conditions were
hypothesized to perform better than the latter. When normalizations were
matrix-matched and bounded, no significant difference for either element
was observed between one-point, two-point, three-point, and four-point
normalizations, although the variance of two-point normalizations is
higher than other methods (Fig. 2A). When the analysis was constrained
to normalizations that were matrix-mixed and extrapolated, the number of
standards used imparted significant differences on the accuracy of the
normalization (Fig. 2B). Two-point N normalizations (median error =
0.232%0, n = 274) had significantly higher error than one-point (median
error = 0.119%0, n = 180, \emph{p} \textless{} 0.0001), three-point
(median error = 0.118%0, n = 278, \emph{p} \textless{} 0.0001), and
four-point normalizations (median error = 0.070, n = 170,
\emph{p}\textless{} 0.0001). Furthermore, three-point normalizations had
significantly higher error than four-point normalizations (\emph{p} =
0.021). Similarly, two-point C normalizations (median = 0.308%0)
exhibited significantly higher error than one-point (median error =
0.209%0, \emph{p} = 0.0005), three-point (median error = 0.187%0,
\emph{p}\textless{} 0.0001), and four-point normalizations (median error
= 0.148, \emph{p} \textless{} 0.0001).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

\textbf{Fig 2} : Boxplots of the accuracy of one-point, two-point, and
multipoint normalizations that are matrix matched and bounded (A) and
matrix mixed/both mixed and extrapolated (B). Significance is shown in
the compact letter display, where any two methods with the same letter
are not significantly different as per Dunn's post-hoc testing. Nitrogen
(uppercase) and carbon (lowercase) letters are not compared with each
other.

\subsection*{Effect of matrix and isotope range on normalization
accuracy}

{\label{effect-of-matrix-and-isotope-range-on-normalization-accuracy}}

To isolate the effect of matrix-matching on normalization errors, we
present results from only three-point normalizations. When the results
are constrained to bounded normalizations, the matrix of the standards
relative to the quality controls has a significant effect for both N and
C (Fig. 3). Nitrogen normalizations that have calibration standards
composed of a different matrix than the quality controls (median error =
0.104%0, n = 688) exhibit significantly higher errors than
matrixed-matched normalizations (median error = 0.064%0, n = 102,\emph{p}
\textless{} 0.0001) and normalizations where the standards and quality
controls are both composed of a combination of matrixes (median error =
0.058%0, n = 190, \emph{p} \textless{} 0.0001). Similarly, mixed-matrix C
normalizations (median error = 0.165%0) exhibit significantly greater
errors than normalizations composed of matched matrixes (median error =
0.056%0, \emph{p} \textless{} 0.0001) and both-mixed matrixes (median
error = 0.121%0, \emph{p} \textless{} 0.0001). Furthermore, N
normalizations with both-mixed matrixes have significantly greater
errors than matched-matrix normalizations (\emph{p} \textless{} 0.0001).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Fig. 3} : Boxplots of the accuracy of bounded three-point
normalizations based on whether the matrixes of the standards and the
quality controls are the same (matched) or mixed. If the three standards
and the two quality controls are both composed of a combination of
matrixes, the normalization is classified as ``both mixed.''
Significance is shown in the compact letter display, where any two
matrixes with the same letter are not significantly different as per
Dunn's post-hoc testing. Nitrogen (uppercase) and carbon (lowercase)
letters are not compared with each other.

To better understand how extrapolating outside of the normalization
curve contributes to normalization errors, we assessed the impact of
extrapolation on three-point normalizations that were matrix-matched
(Fig. 4). Three-point normalizations that had at least one quality
control within the isotope range of the normalization (i.e.,
``bounded'', n = 102) had decreased normalization errors for N and C
(median error = 0.064%0 and 0.056%0, respectively) compared to
extrapolated normalizations (n = 30) for N (median error =
0.116%0,\emph{p} \textless{} 0.0001) and C (median error = 0.132%0,
\emph{p}\textless{} 0.0001).The isotope ranges of the two-point and
three-point normalizations were then binned into three categories for
statistical assessment: less than 15%0, 15%0 to 30%0, and greater than 30%0.
When only bounded matrix-matched normalizations are considered, neither
two-point or three-point normalizations exhibit a significant
relationship between isotope range and normalization error (Fig. 5A,
5B), with median normalization errors less than 0.1%0. However, when the
data is constrained to extrapolated matrix-mixed normalizations,
normalizations with a range less than 15%0 have significantly greater
errors for both two-point and three-point normalizations (Fig. 5C, 5D).
Under these conditions, three-point N normalizations with an isotope
range less than 15%0 had significantly higher normalization errors
(median error = 0.171%0, n = 170) compared to those with a range of 15%0
to 30%0 (median error = 0.093%0, n = 46, \emph{p} = 0.0109) and those with
a range of 30%0 to 45%0 (median error = 0.075%0, n = 62, \emph{p}
\textless{} 0.0001). Similarly, three-point C normalizations with an
isotope range less than 15%0 had significantly higher normalization
errors (median error = 0.219%0, n = 186) than those with a range of 15%0
to 30%0 (median error = 0.147%0, n = 92, \emph{p} = 0.0045). Two-point
normalizations that were extrapolated and matrix-mixed also fared worse:
N normalizations with a range less than 15%0 exhibited significantly
higher normalization errors (median error = 0.262%0, n = 200) than 15%0 to
30%0 (median error = 0.0890%0, n = 56, \emph{p} = 0.0023) and those with a
range of 30%0 to 45%0 (median error = 0.0923%0, n = 18, \emph{p} = 0.0155),
and C normalizations with an isotope range less than 15%0 had
significantly higher errors (median error = 0.360%0, n=248) than those
with a range between 15%0 and 30%0 (median error = 0.107%0, n = 26,
\emph{p} \textless{} 0.0001). No C normalizations that were extrapolated
and matrix-mixed had a range greater than 30%0.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

\textbf{Fig. 4} : Boxplots of the accuracy of matrix-matched three-point
normalizations based on whether at least one of the quality control
standards falls within the isotope range of the standards (bounded) or
if both quality control standards fall outside the isotope range of the
standards (extrapolated). Significance is shown in the compact letter
display, where any two extrapolation status with the same letter are not
significantly different as per Dunn's post-hoc testing. Nitrogen
(uppercase) and carbon (lowercase) letters are not compared with each
other.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

\textbf{Fig. 5} : Normalization error of multipoint normalizations based
on the isotope range of the standards. A: matrix matched and bounded
three-point normalizations do not exhibit significant differences with
isotope range; B: matrix matched and bounded two-point normalizations do
not exhibit significant differences with isotope range; C: matrix mixed
and extrapolated three-point normalizations have significantly higher
normalization errors when the isotope range is less than 15%0; D: matrix
mixed and extrapolated two-point normalizations have significantly
higher normalization errors when the isotope range is less than 15%0.
Significance is shown in the compact letter display, where any two
isotope ranges with the same letter are not significantly different as
per Dunn's post-hoc testing. Nitrogen (uppercase) and carbon (lowercase)
letters are not compared with each other.

\subsection*{Interlaboratory comparison of instrument
linearity}

{\label{interlaboratory-comparison-of-instrument-linearity}}

To assess the linearity effect across laboratories and instrument
manufacturers, five different working standards were each analyzed
across a range of sample weights (n=10) while the sample dilution was
held constant, thus producing a range of peak amplitudes. Both
instruments exhibited a non-linear deviation in reported\selectlanguage{greek}\emph{δ}
\selectlanguage{english}\textsuperscript{15}N as beam amplitude decreased (Fig. 6) -- at U.S.
EPA this deviation occurred at amplitudes below 6nA, and at UNM CSI it
occurred below 2V (peak amplitude is measured in units of current (nA)
on Elementar instruments and voltage (V) on ThermoFinnigan instruments).
The linearity effect of reported\selectlanguage{greek}\emph{δ} \selectlanguage{english}\textsuperscript{15}N occurred
regardless of sample matrix and resulted in substantial
(\textgreater{}1.5%0) decrease from the median\selectlanguage{greek}\emph{δ}
\selectlanguage{english}\textsuperscript{15}N at both facilities. Instrument linearity effects
for \selectlanguage{greek}\emph{δ} \selectlanguage{english}\textsuperscript{13}C were not evident at UNM CSI across
the observed range of peak amplitudes. At U.S. EPA, instrument linearity
effects for \selectlanguage{greek}\emph{δ} \selectlanguage{english}\textsuperscript{13}C were not observed below 20nA,
but linearity effects were observed for beam amplitudes above 20nA (Fig.
6).

Within 48 hours of analyzing the solid samples, diagnostic reference gas
linearity tests were performed for N and C at U.S. EPA and for N at UNM
CSI with the same tuning and configuration. Both facilities displayed a
small reference gas linearity effect for N across the tested range of
peak amplitudes (Fig. 6), with a total isotope range of 0.20%0 and 0.27%0
at UNM CSI and U.S. EPA, respectively. At the U.S. EPA, the reference
gas C linearity diagnostics suggested a consistent inverse relationship
between reported \selectlanguage{greek}\emph{δ} \selectlanguage{english}\textsuperscript{13}C and beam height across
the tested range of peak amplitudes (2-12nA), with a total isotope range
of 0.33%0.

To investigate whether the amount of combustion in the elemental
analyzer varies as a function of sample weight, and thus influences the
linearity results, we assessed how the ratio between peak amplitude and
sample weight varied as a function of sample weight (Figure S1).
Combustion effects appeared to be matrix-dependent, with high-organic
matrixes exhibiting a higher variation in the ratio between peak
amplitude and sample weight than other matrixes.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

\textbf{Fig. 6} : Linearity effects across sample matrixes at the U.S.
EPA (left) and UNM CSI (right). The linearity effect is shown relative
to the median observed isotope composition of each working standard.
Vertical red lines indicate nominal peak amplitudes that were targeted
for tuning and sample peak heights. Note that units of peak amplitude
vary between instruments.

\subsection*{Discussion}

{\label{discussion}}

\begin{enumerate}
\item ~
  \subsection*{Two-point isotope normalizations are insufficient for
  EAIRMS}

  {\label{two-point-isotope-normalizations-are-insufficient-for-eairms}}
\end{enumerate}

In this study, we conducted a total of 6272 normalizations for N and C
at two laboratories using 8 certified isotope reference materials. Past
work has found that one-point normalizations have larger normalization
errors than two-point, three-point, and four-point
normalizations\textsuperscript{11,17,27}, and our results further
suggest that normalization accuracy generally improves with the number
of standards. Regardless of whether we test bounded, matrix-matched
normalizations or extrapolated, matrix-mixed normalizations, three-point
and four-point normalizations exhibit better accuracy than one-point and
two-point normalizations (Fig. 2). The lowest range of normalization
errors was consistently found for three-point and four-point
normalizations, when samples were bounded within the range of
calibration standards, and when the sample matrix matched between
samples and calibration standards (Fig 2A).

The dramatic reduction in two-point normalization accuracy when the
normalization was extrapolated and matrix mixed (Fig. 2B) was surprising
given that foundational literature suggests that two-point
normalizations are sufficient for the normalization of stable isotope
results\textsuperscript{17,21,37}. Extrapolating beyond the
normalization and mixing matrixes between the samples and standards are
expected to increase normalization errors, regardless of how many
standards are used. However, the median error of two-point
normalizations conducted under these abnormal conditions were 64\%-230\%
greater than corresponding three-point and four-point normalizations,
with the error of some two-point normalizations exceeding 1%0.

The sensitivity of two-point normalizations to matrix effects and
extrapolation are evident in its derivation process (Eq 3-5). In a
two-point normalization, the sensitivity of \emph{m} to inaccuracies
in\selectlanguage{greek}\emph{δ\textsubscript{raw(std)}}\selectlanguage{english} is a function of the isotope range of
the two standards -- as the isotope range of the standards
decreases,\emph{m} becomes more sensitive to isotope variability of the
standards\textsuperscript{21}, including those due to matrix effects.
Indeed, we find that two-point normalizations have the highest
\emph{m}variability (Fig. 7A), and that the variability of \emph{m} is
inversely related to the isotope range of the standards (Fig. 7B). The
effect of\emph{m} on \selectlanguage{greek}\emph{δ\textsubscript{true(sample)}}\selectlanguage{english} also varies as
a function of the isotope range between the standards and the samples:
as the difference between \selectlanguage{greek}\emph{δ\textsubscript{raw(std)}}\selectlanguage{english}
and\selectlanguage{greek}\emph{δ\textsubscript{raw(sample)}}\selectlanguage{english} increases, errors due to an
incorrect \emph{m} are compounded. Changes
in\selectlanguage{greek}\emph{δ\textsubscript{raw(std)}}\selectlanguage{english} due to matrix effects will manifest
as inaccuracy when the matrix of the standards and the samples are
mixed, which would subsequently be amplified by the effects of
extrapolation. Thus, two-point normalizations that are extrapolated and
matrix-mixed have poor performance -- particularly with a small isotope
range between the two standards (Fig. 5D). Overall, this work questions
whether two-point linear normalizations are sufficient for biological
applications of EAIRMS -- in our study, many two-point normalizations
were inferior to one-point normalizations because the actual \emph{m}was
close to 1, the assumed slope for 1-point normalization. Notably,
three-point and four-point normalizations are much more resilient to the
effects of extrapolation and matrix mixing (Fig. 2B), even when the
isotope range was less than 15%0 (Fig. 5C). Thus, we recommend that all
users of EAIRMS systems use at least 3 calibration standards to compose
their normalization curve.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}

\textbf{Fig. 7} : The slope of all multipoint normalizations by isotope
range of the standards. Dashed horizontal lines indicate the slopes for
each facility derived from an 8-point normalization composed of all
certified reference standards.

\subsection*{Normalizations are significantly impacted by matrix and
extrapolation
effects}

{\label{normalizations-are-significantly-impacted-by-matrix-and-extrapolation-effects}}

Even when results are constrained to three-point normalizations, the
matrix of the standard relative to the quality controls has a
significant effect on the accuracy of the normalization (Fig. 3).
Bounded normalizations where the matrixes of the standards were mixed
relative to the quality controls exhibited median errors 63\%-195\%
greater than those where the matrixes were matched. The mechanism behind
this matrix effect is unclear, particularly because the isotope
composition of the plant-based standards were quantified using high
organic certified reference materials\textsuperscript{20}. In accordance
with typical EAIRMS usage, the plant-based standards were weighed to a
higher mass and analyzed with a higher oxygen dosing and a greater
sample dilution for C, and thus it is possible that these
instrumentation factors are contributing to the matrix effect observed
in this study. Although past work has suggested that matrix matching
between organic and inorganic samples would reduce normalization
errors\textsuperscript{28}, this study posits that matrix effects should
be considered even within organic samples. In studies where
matrix-matched certified reference materials do not exist (e.g.,
sediment; glass fiber filters), the effects of matrix-mixing should be
considered as a source of imprecision that will not be reflected in the
variance of the standards alone.

Similarly, this study shows that extrapolating beyond the isotope range
of the standards-- not an uncommon occurrence when analyzing a wide
variety of biological samples -- increases median normalization errors
by 81\%-135\%, even when the analysis is constrained to matrix-matched
normalizations (Fig. 4). Normalizations with a smaller isotope range are
more likely to require extrapolation, but the lack of significant
relationship between isotope range and normalization error for bounded,
matrix-matched, three-point normalizations (Fig. 5) indicates that
extrapolation is the primary factor driving inaccuracy. Overall, our
results provide experimental evidence to support the emerging
consensus\textsuperscript{37} that calibration standards for EAIRMS
should have isotope values that span the full natural range of the
measured elements, regardless of the isotope values of the samples being
analyzed.

\subsection*{Instrument linearity cannot be predicted by reference gas
diagnostics
alone.}

{\label{instrument-linearity-cannot-be-predicted-by-reference-gas-diagnostics-alone.}}

Although reference gas linearity diagnostics are anecdotally used as a
means of assessing instrument performance, we quantified instrument
linearity using replicate analyses of working standards across a range
of sample weights. At both facilities, a large linearity effect was
observed in the reported isotope composition of the working standards
that was not reflected by reference gas linearity diagnostics, which
inject pulses of reference gas into the IRMS to assess the linearity
effect. The linearity effect does not correspond to incomplete
combustion in the elemental analyzer (Fig. S1), suggesting that other
factors, such as a N blank introduced during sample preparation, are
driving our observations. These results show that, while reference gas
linearity diagnostics may be useful for confirming normal instrument
operation, they are not representative of the linearity effect that will
be observed when analyzing solid samples. We suggest running replicate
measurements of a solid standard at varying sample weights to determine
the peak amplitudes at which linearity has an effect and applying a
linearity correction curve when necessary. In our study, N linearity was
observed at peak amplitudes corresponding to \textasciitilde{}0.06 mg of
N, meaning that instrument linearity is of particular importance for
samples with a low N content such as sediment and plant tissue. The high
magnitude of the linearity effect observed in both facilities across a
range of sample matrixes reinforces the importance of this instrument
effect for any facility measuring stable isotopes in biological samples.

\subsection*{Conclusion -- best practices for biological applications of
EAIRMS}

{\label{conclusion-best-practices-for-biological-applications-of-eairms}}

Through an experimental assessment of isotope normalizations across a
variety of sample matrixes and isotope ranges, we assessed how the
number, matrix, and isotope range of the calibration standards effected
normalization error on two EAIRMS systems. In the first known assessment
of normalization methods for both N and C, we found that three-point and
four-point normalizations have the lowest normalization errors and are
most resilient to the effects of sample matrix effects and
extrapolation.

Past work has identified two-point normalizations as a common and
acceptable means of normalizing isotope results\textsuperscript{2,17}.
In contrast, some of the observed two-point normalizations had
deviations more than 1%0 and were significantly less accurate than
three-point and four-point normalizations. We posit that two-point
normalizations are vulnerable to the effects of matrix-mixing and
extrapolation and should not be considered sufficient for EAIRMS
normalization in biological applications. Although normalizations using
at least three standards were more resilient to these factors, we found
that mixing the matrix between the samples and the standards and
extrapolating outside of the curve reduced accuracy regardless of how
many standards were analyzed. No significant impact was observed with
the isotope range of three-point normalization but maximizing the
isotope range will reduce the likelihood of extrapolation. Thus, we
recommend users of EAIRMS systems normalize their results using at least
three calibration standards that span a large isotope range and are
matrix matched with the samples being analyzed, and to include at least
one additional independent quality control standard.

In our interlaboratory comparison of instrument linearity, we found that
linearity error was substantial, especially for N, regardless of
instrument or sample matrix. Diagnostic reference gas linearity testing
was unable to reproduce the observed linearity effect, suggesting that
the reference gas is not a useful predictor of real-world instrument
linearity. Although reference gas diagnostic testing may be a beneficial
tool for assessing nominal instrument operation, the linearity response
of the instrument should be assessed experimentally by analyzing solid
standards across a range of masses.

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