Four conceptual categories contributing to HR-AMS uncertainties can be defined: interpretation, counting, instrumental, and analysis uncertainties. These uncertainties may be defined and addressed as follows.

Interpretation uncertainties arise when a given signal may arise from sources other than the analyte. For example, in the AMS, H2O+ ions may form when either gas-phase water, particle-bound water, or the thermal decomposition of oxygenated functional groups in organic PM (OM) , inorganic PM , or BC . The interpretation of an H2O+ ion as originating from any one of these sources therefore requires additional information or assumptions to be made. In the AMS, a fraction of the overall H2O+ signal is typically attributed to OM according to laboratory-measured fragmentation patterns as detailed by . This so-called fragmentation-table approach may introduce biases but not imprecision to the data set. The biases would be constant only for a constant aerosol composition.

Interpretation uncertainties may also occur if an ion is misidentified or omitted in the PIKA software. Such errors are not considered in this work.

In PIKA (up to the current version, 1.11C), the signal intensity y due to a specific ion is considered to arise from a peak ϕ, together with some noise σy, for each measured ion-time-of-flight x (corresponding to different m/z): y(x)=ϕ(x)+σy.

To a first approximation, the peak ϕ has a Gaussian shape and may be modelled by G=h⋅exp⁡-(x-μ)2w2, where h is the peak amplitude (peak height), μ the mode and mean of the Gaussian (peak location), and w the standard deviation (SD; peak width). These parameters may be estimated by least-squares fitting to the data.

In practice, the Gaussian model is modified to account for peak broadening, skewness, tailing, or other instrumental non-idealities by defining a peak-shape function v. The v is determined empirically by averaging a large number of normalized, well-resolved peaks from a given data set and may be visualized as a one-dimensional vector of correction factors in normalized x space (xnorm.=[x-μ]/w). Including v in Eq. () defines the pseudo-Gaussian function , f=v⋅G=h⋅v⋅exp⁡-(x-μ)2w2,f=h⋅f0.

To improve the robustness of the PIKA fitting routine against poorly resolved peaks and noisy data, some of the parameters in Eq. () are constrained during peak fitting. The v is determined as described above, μ is obtained from an m/z calibration, and w is from a peak-width parameterization, w(m/z). These are obtained as follows.

The m/z calibration is achieved by fitting Eq. () to user-selected background peaks. The fitted peak locations μ in time-of-flight space are calibrated to their known m/z following the procedure described by . Separate m/z calibrations are obtained for every mass spectrum. In practice, these calibrations are applied to predict peak locations in ion time-of-flight space before fitting the raw data. We therefore refer below to “μ-prediction errors” rather than “m/z-calibration errors” except when specifically discussing m/z calibrations. Although all peaks were fitted in ion-time-of-flight space below, the discussion refers to peak m/z for clarity.

The w(m/z) calibration is also obtained by fitting Eq. (), but to a separate set of peaks. This set of peaks is carefully selected to ensure that only single, well-resolved peaks of unambiguous composition are considered. The fitted peak widths w are averaged before being parameterized by a (typically linear) calibration curve, as described by . In the present analysis, the w(m/z)-calibration procedure was slightly modified to improve its robustness against poor-quality peak fits, as described in detail in and briefly in the Appendix.

Thus, the three inputs to Eq. () are defined from three different calibrations on the data. These calibrations are fundamentally different: the w(m/z) and v calibrations are determined once for the entire data set, while the m/z calibration is determined for each individual mass spectrum. Variability in the accuracy of the m/z calibration may therefore lead to variability in the analysis. Also, different peaks are used for all three calibrations, so that their prediction errors may also differ.

With the inputs described above, Eq. () is fitted to estimate h. An example is shown in Fig. , showing the CO2+ signal at m/z 44 in the lower panel and the fit residuals in the upper panel. The peak spans only 2–3 detector bins (corresponding to 4–6 ns), which was typical for m/z in the range 15–75 for this specific instrument. Other studies have reported slightly lower m/z resolutions and correspondingly a slighly higher number of detector bins representing each peak e.g.. The conclusions drawn at the end of this study are independent of this detail.

The fitted h is used by PIKA to estimate the area A of each peak via the Gaussian integral as A(G)=hwπ, which leads to A=hwπ⋅Af0AG0⋅kDC, where AG0 and Af0 are the respective integrals of the standard Gaussian and pseudo-Gaussian distributions. These standard integrals are defined by Eqs. () and () with unit h and w, and μ=0. The factor kDC is a correction for the mass-spectrometer duty cycle to account for the fact that lighter ions will be transported faster than slower ions into the mass spectrometer prior to time-of-flight measurement and, consequently, reside in the extraction region for shorter times . The ratio Af0/AG0 is typically close to unity.

After estimating A, PIKA currently estimates the uncertainty in A from the square root of the number of ions associated with the signal, according to an assumed Poisson distribution of errors .

In the HR-AMS, more than one peak is typically observed at each integer m/z. These peaks are modelled as a linear superposition fT of pseudo-Gaussians with similar w : fT=∑i=1jhif0,i(μi,w).

The uncertainty of fits to Eq. () with j>1 (multiple overlapping peaks) is much more complex than the case where j=1 (≡ well-resolved peaks).

The next section therefore focusses on the case of well-resolved peaks, before Sect.  discusses the overlapping-peak case.

The integration of fits to well-resolved pseudo-Gaussian peaks via Eq. () will lead to an uncertainty, σAA2=σhh2+σww2, which, if the fractions σh/h and/or σw/w are sufficiently large, may contribute significantly to the overall uncertainty of integrated mass-spectral signals.

In Eq. (), the uncertainties in Af0 and kDC are considered negligible, since the integral AG0 is known exactly; the integral Af0 is also known exactly for a given v; and the uncertainty in kDC depends only on the stability of the mass-spectrometer voltages, which fluctuate negligibly for the time periods over which data are collected (Sect. ). In addition, h and w have negligible covariance as they are determined via separate and independent calibrations (Sect. ); the w(m/z) calibration requires, and can be used to validate, the assumption that w is independent of h.

The value of σw may be directly estimated from the peak-width calibration procedure via the uncertainties in the calibration-fit coefficients. For the test data set discussed below, this approach indicated σw/w≈2.5%. Such a percentage uncertainty is important as it scales with signal differently to the typically used Poisson uncertainty, increasing linearly with the number of ion counts rather than as the square root.

The estimation of σh is much more complex than σw. Whereas σw may be estimated directly from the calibration fit, σh reflects the imprecision of the constrained PIKA fitting procedure (Sect. ), the constraints of which are predicted by two separate calibrations (peak width and location) as well as an empirically defined peak shape v.

In this subsection we describe the theoretical basis on which the RMSE, defined below, has been interpreted in Sects.  and . The focus is on well-resolved peaks; overlapping peaks are addressed in Section .

Since the width, location, and shape of the PIKA pseudo-Gaussian fit function (Eq. ) are predefined by different calibrations, the exponential term of Eq. () reduces to a transformation of the x variable: f=hf0(x,w,v,μ), where f0 is defined by Eq. () with h=1 and with μ, w, and v given by the calibrations described in Sect. .

A fit to Eq. () is equivalent to a simple linear regression. (Indeed, the newest version of PIKA directly performs such a regression.) An uncertainty estimate for h could thus be obtained directly from the regression, but this estimate would not account for bias in the fit. Such bias must be accounted for, because when such bias varies from fit to fit (i.e. from mass spectrum to mass spectrum for the same peak), it produces an imprecision in h. Such a varying bias is to be expected, because a new f0 is defined for each fitted ion. The accuracy of f0 may vary in this context due, for example, to variations in the accuracy of the m/z-calibration equation which is obtained for each mass spectrum.

The influence of such variations on the imprecision of the fitted h values cannot be estimated directly from a single fit but must be considered in the context of the whole data set. It can be expected, however, that (in the absence of noise) errors in h due to f0 will be proportional (scale linearly with h) since f0 is scaled by h during fitting.

To quantify the error in a given fit of f, we use the SD of the fit residuals, the fit root-mean-square error (RMSE): RMSE=1j-1∑i=1j[y(xi)-f(xi)]2=1j-1∑i=1j[ri]2=SSRj-1.

As the RMSE of a single peak is the main quantity of interest below, the range 1≤i≤j was limited to 2 SDs on either side of the peak centre (μ±2w), representing the trimmed variance of the residuals e.g..

The RMSE is not used to directly infer the error in the fitted h but rather as a diagnostic with which the causes of errors in f0 may be elucidated and understood. Based on this understanding, the imprecision in h is then estimated.

For a sufficiently large sample, the expected value of the squared RMSE is the sum of the model variance and the squared model bias . In the context of this discussion, this statement may be illustrated as follows. The RMSE represents the expected value of a fit residual, 〈ri〉, which is related to the fitted and true peak functions hfitted,if0 and htrue,iϕ0 as 〈ri〉=〈hfitted,if0-htrue,iϕ0+σy,i〉, where hfitted,i and htrue,i are scalars to the estimated and true unit-height peak functions f0 and ϕ0 respectively, and σy,i represents noise in the data due to ion counting or background subtraction errors.

Letting f0,i=ϕ0,i+εi, where εi is the error in f0 at each data point resulting from imperfect knowledge of the true peak shape, 〈ri〉=〈hfitted,iϕ0+hfitted,iεi-htrue,iϕ0+σy,i〉=hfitted,iϕ0+hfitted,i〈εi〉-htrue,iϕ0+〈σy,i〉.

For small peaks, counting uncertainties dominate the fit residual: (htrue,iϕ0)/σy,i⟶0, thus 〈ri〉⟶σy,i in Eq.  (cf. Fig. ).

For large peaks, the counting uncertainties σy,i become negligible relative to hfitted,i〈ε〉. In this situation, the slope of a plot of 〈ri〉 against hfitted,i will be

d〈ri〉dhfitted,i=〈εi〉-ϕ0,idhtrue,idhfitted,i-1=〈εi〉, where dhtrue,i/dhfitted,i=0 if the peak shape is independent of the peak intensity, as was already assumed (Sect. ). This assumption is discussed further in Sect. .

Thus, for signals large enough for noise to be negligible, a plot of 〈ri〉 (or an estimator of 〈ri〉, such as the RMSE) against the fitted peak height hfitted may show a linear slope where errors in the scaled peak-shape function f0 dominate counting errors. The linearity arises because of the linear scaling of f0 by hf. If a unique f0 is prescribed for each fit, as is the case in PIKA, then variability within the set of prescribed f0 may lead to an overall imprecision σh in the set of fitted hfitted. Below, we refer to hfitted simply as h. For fits to well-resolved ions, we plot the RMSE against h to evaluate the factors that lead to the constant relative error defined in Eq. () in order to build an understanding on which the corresponding σh may be estimated in Sect. .

The RMSE of a number of well-resolved peaks in the test data set are plotted in Fig. . Each point in the figure represents the RMSE of a fit of Eq. () to the data. The figure includes seven well-resolved peaks based on their being both well-resolved from any neighbours and present at a wide range of signal intensities. The mass spectra have been background subtracted before fitting as described in the Appendix. Background subtraction was performed to remove inconsequential differences between the ions, for example due to signals from background CO2 gas. Other ions had very small background signals regardless.

The peaks in Fig.  span a range of different m/z, as indicated by the integer m/z values shown in the legend, and represent a range of different species. For example, C3+ ions were formed after BC vaporized at ∼ 4000 K, whereas C2H3O+ and C3H7+ ions were formed after organic PM vaporized below 873 K (600 ∘C) . In contrast to all other ions, these latter two organic ions are found at the same integer m/z and were therefore fitted simultaneously as a linear sum in PIKA (Eq.  with j=2). This point is returned to in Sect. .

Two distinct trends are evident in Fig. . First, the RMSE approaches an asymptotic value for low peak heights (∼1 counts s-1 bin-1; cps bin-1). This approximately constant RMSE (σy≈0.2 cps bin-1) is representative of noise in the data y. Second, the RMSE shows a constant slope for high-signal peaks, indicating an approximately constant relative RMSE for each peak (RMSE/h≈3%). This illustrates the importance of fitting-function errors during the fit (Sect. ). The constant RMSE and constant relative RMSE are respectively depicted by the dotted and dashed lines in Fig. , as well as their quadratic sum.

The constant RMSE/h for high signals suggests that σh/h may also be constant. Thus σh may also scale linearly with signal, as was the case for σw/w (Sect. ), and may significantly influence the overall peak-integration uncertainty for high signals. The next sections therefore aim to obtain an understanding and quantification of σh/h for incorporation into the overall HR-AMS uncertainty.

The source of the constant relative RMSE in Fig.  was argued via Eq. () to be most likely due to errors in the predefined pseudo-Gaussian function f0. To elucidate which of the parameters defining f0 have the greatest impact on the errors of the resulting fit, imprecisions or biases were manually added to each of several input parameters in the PIKA analysis procedure as follows.

The original magnitude of each error was estimated directly from the data. Based on these estimates, significantly larger uncertainties were added to the data, as specified in Table . In most cases, the errors specified in the table were used to define the SD of a Gaussian probability distribution from which a new error was sampled for each analysed peak.

Figure  plots the effects of these errors on the resulting RMSE for one exemplary ion, C3H7+. This ion was chosen simply because it represented a large range of peak intensities (abscissas in Fig. ); the trends seen in Fig.  were observed for all of the ions in Fig.  (not shown). The magnitudes of the errors used to generate the figure are highlighted in boldface in Table  and were chosen so as to give a visible change in the RMSE graphs (where possible), not to represent realistic errors.

With this approach, multiple potential sources of the constant relative RMSE term can be eliminated: noise in the predicted peak width w (Fig. b), errors in the slope of the subtracted linear baseline (Fig. d), and the use of a different AMS vaporizer (Fig. e) did not alter the magnitude of the relative RMSE.

Conversely, two potential sources of the constant relative RMSE can be identified: noise in the predicted peak location μ (Fig. f) and errors in the peak-shape v (Fig. g and h). In these three cases (Fig. f–h), a relative RMSE of ∼15 % was observed.

The relative μ-prediction errors necessary to achieve this ∼ 15% relative RMSE were 5–10-fold larger than typical μ-prediction errors; the errors in v necessary to achieve this ∼ 15% relative RMSE were 2 orders of magnitude larger than the maximum distance between the best estimate of v and the data used to derive it, which was 0.4%. This suggests that m/z-calibration uncertainties play a larger role than peak-shape uncertainties in a real data set. This difference in sensitivity is not surprising given that μ is part of the exponential term in Eq. () while v is a linear term. In the next section, the fitting procedure was altered to prove that the majority of the fitting error was due to μ-prediction errors.

One additional implication of Fig. g and h is that RMSE/h is a function of v. Given that Fig.  showed that this fraction was nearly constant for high signals, we infer that the peak shape was nearly constant across this range of signals. If v was independent of signal intensity, then the individual ions comprising each peak interacted negligibly within the instrument, which provides justification for the assumed behaviour of overlapping peaks as a linear sum in Eq. ().

To explore the impact of errors in μ prediction on the fit RMSE, the fitting procedure was altered to allow μ and/or w to be varied during fitting. The peak shape v could not be meaningfully varied.

Allowing μ to vary by ±20 ppm during fitting reduced the fit RMSE by almost 1 order of magnitude at high signals (Fig. , red triangles), indicating that the majority of the fitting error was due to errors in the predicted μ (which result from both imprecisions and biases in m/z calibration). The RMSE at low signals was virtually unchanged, since it was dominated by noise in the data (Sect. ). While Fig.  illustrates this result using C3H7+, these conclusions were verified for all of the well-resolved ions discussed above.

Further relaxing the μ constraint to ±50 ppm did not further reduce the RMSE. Conversely, reducing the constraint to ±10 ppm resulted in a change intermediate between the 0 ppm (fully constrained) and ±20 ppm constraints, suggesting that the true accuracy of the m/z calibration was on the order of ±15 ppm. This value does not correspond to the imprecision of the m/z calibration, as discussed in the next section.

Allowing w to vary during the fit by ±5 % (2σw) had no observable effect on the RMSE (Fig. ). Although this result may appear to contradict the fact that a significant uncertainty in the w calibration was estimated during calibration (Sect. ), it is fully consistent with the only minor changes in RMSE observed when large errors were added to w in Sect.  (Fig. b). Thus, although uncertainties in w do not strongly influence the fitted h, they lead to uncertainties during the peak integration by Eq. () (cf. Eq. ).

When both w and μ were allowed to vary during fitting, the RMSE behaved similarly to the case where only μ was allowed to vary (i.e. the red triangles in Fig. ). In this case, some linear dependence of the RMSE on the fitted peak height remained, suggesting that f0 still had some influence on the RMSE. This remaining dependence is attributed to errors in the peak-shape factor v. Since the empirically defined v (a vector of more than 100 values) could not be allowed to vary during fitting as μ and w were in Fig. , its influence on the RMSE trends could not be explored in this context. If the remaining RMSE trend was due to errors in v, these errors had a much-smaller impact on the RMSE (and therefore the goodness of fit of Eq. ) than the errors in μ prediction, so they are not discussed further.

When the predicted peak location μpredicted was allowed to vary during fitting (Sect. ) the relative RMSE was significantly reduced in the high-signal regime. The final fit therefore represented a significantly better model of the data, which is interpreted as an improvement due to a reduced error in the μ. The fitted peak location, μfitted, is therefore interpreted as a good approximation to the true peak location, μtrue, and used to estimate errors in μ prediction as eμ≈(μfitted-μpredicted) in the current section. Limitations of this estimation approach are discussed in Sect. .

The dark-shaded data in Fig.  show the distribution of the estimated μ-prediction errors for peaks outside of the noise regime, defined as hfitted>20 cps/bin from Fig. . For this analysis, μ was allowed to vary by ±40 ppm. If the aggregated eμ data are fitted by a Gaussian function, the mean appears to be virtually 0 (fitted value: -0.5±0.8 ppm), suggesting zero bias in μpredicted. However, the bias for each individual ion was not 0, as shown by the light-shaded data in the same figure. The μfitted for each ion showed significant biases (mean different from 0) as well as imprecisions (significant spread relative to the mean). The magnitude of these biases and imprecisions were estimated by a Gaussian fit in each case, with the fitted values shown in Table . The imprecisions in μpredicted are generally of similar magnitude to the biases, and the biases show both positive and negative values. These biases must therefore be taken into account when estimating h-fitting imprecision for individual ions.

Figure  plots the data for m/z 43, C2H3O+ (m/z 43.0184), and C3H7+ (m/z 43.0548) that were shown in Fig.  as a scatter plot. As these two ions are of similar m/z, but are nonetheless well resolved with a separation of ∼7 standard deviations cf. , it was hypothesized that their μ-prediction biases may have been approximately equal. Such approximate equality would have allowed μ-prediction biases to be fitted as a single free parameter during fitting of Eq. (). Alternatively, a single value for the μ-prediction bias could have been assumed during Monte Carlo analysis (Sect. ).

The lack of a 1:1 trend in Fig.  (indeed, a slight negative trend is evident) suggests that the μ-prediction biases for nearby peaks cannot be assumed to be equal. This unexpected result may be related to the sensitivity of the fits to the location of the mode of a peak relative to the nearest detector bin, which is discussed in Sect. . The method by which eμ was estimated may also play a role, but to some degree this method must also represent the sensitivity of the fitting algorithm to the data. Since this sensitivity also influences the ultimate quantity of interest, σh, this conclusion would remain true regardless of the accuracy and precision of the eμ estimate. Moreover, we emphasize that ion-dependent biases in m/z calibration are not unique to this study, as discussed in Sect. .

Although it is readily apparent that a μ-prediction bias may lead to a bias in the fitted h for a given ion, it is less apparent that such a bias also significantly increases σh for any given μ-prediction imprecision. This occurs because the slope of a peak increases with increasing distance from its mode (for small distances), such that the variation in fitted height is larger for peaks fitted further away from their mode. The magnitude of this effect is illustrated in Fig. , which is discussed below. It is therefore essential consider μ-prediction biases when estimating the magnitude of σh.

The blue points in Fig.  (case A) show the σh corresponding to the RMSEs shown in Fig. . The imprecision increases as h becomes smaller due to the noise in the data but reaches an asymptotic value as h becomes larger. As noted above, these points represent random samples from the entire test data set.

In practice, it is desirable to estimate σh on a case-by-case basis for each peak in each mass spectrum. Such an estimate is shown by case B in Fig. , which was obtained by performing the Monte Carlo approach described above except the random selection of different m/z axes was not performed. Different m/z axes showed different asymptotic values; the selected axis was chosen for its visual distinction. This difference does not represent real variability in the data: again, only the m/z axis was taken from the data set and the peak itself was simulated. Rather, the difference was due to a shift in the relative distance between the data points representing the peak and the m/z axis (the “bin-to-mode distance”), which is arbitrary and can vary between mass spectra. Case C demonstrates this fact by repeating case B while shifting the mode location of the peak by a randomly selected fraction of the bin spacing between 0 and 1 for each Monte Carlo fit. The result is an estimate of σh that is comparable to the entire-data-set case and represents fit-function errors rather than the bin-to-mode distance.

For overlapping peaks, the μ-prediction imprecision and bias is generally not known. Case D repeats case C but without the μ-prediction bias. The result is an imprecision estimate that is much lower. For case E, a much higher imprecision of 20ppm has been used, based on the recommendation from Sect.  that the estimated μ-prediction imprecision be increased to allow unbiased estimation of σh when μ-prediction biases are unknown.

In Fig. , the standard deviation of multiple fits was used to represent σh. However, the distribution of errors comprising σh was not normally distributed, as shown in Fig. . Figure  includes all seven of the well-resolved peaks discussed above and only represents signals much larger than the noise level, since each followed the same asymptotic behaviour shown in Fig.  and since counting errors must ultimately be combined with peak-integration errors at low signal-to-noise ratios (Fig. ). A large number of iterations (5000) were performed for each peak so that histograms could be plotted in detail. The mean fitting error (bias) has been subtracted from each distribution so that the plot represents σh. The errors are normalized to the absolute peak height, as they were a constant fraction of the height (as expected from Sect. ). For all ions, the limiting case of very high ion signals was simulated. The reader is referred to for a discussion of the effects of ion-counting uncertainties on fitting imprecision.

Each probability distribution in Fig.  shows a very different skewness. These differences are largely controlled by the μ-prediction bias and imprecision, which may cause the predicted peak location to be consistently away from the peak mode. Away from the mode, the peak itself slopes more steeply and a small μ-prediction imprecision leads to a large variability in the fitted height. This fact means that a greater imprecision results from a greater bias in the m/z calibration. The μ-prediction bias is therefore an important component of the imprecision of the fitted h. Without including this bias in estimating σh, the resulting imprecision estimates would be considerably different (Table ).

The impact of the μ-prediction biases discussed above on σh will be reduced for broader (lower m/z-resolution) peaks, since such peaks are less steeply sloped. Broader peaks are also represented by more points in the mass spectrum, which has a major impact on the imprecision: for the case of C3H7+, increasing the peak width by a factor of 1.5 (which doubled the number of data points representing >1 % of the peak height from 4 to 8) decreased the imprecision from 1.6 to 0.2 % (Table ). This effect would not necessarily be reduced by increasing the number of points per peak without changing the mass-spectral resolution but instead only by reducing the μ-prediction errors.

A non-negligible m/z-calibration bias is anticipated for any given mass-spectral peak, since ion trajectories are typically imperfectly described by the calibration function as discussed in Sects.  and . We explored the sensitivity of σh to this bias by calculating σh for the three exemplary peaks shown in the top row of Fig. , for a wide range of μ-prediction bias and imprecisions. Figure  thus illustrates the range of imprecision for which a given m/z-calibration bias becomes negligible: for the range of m/z-calibration biases where σh remains similar along a vertical cross-section of the middle panels of the figure, the actual value of the bias term need not be known.

To place Fig.  in context, the μ-prediction biases and imprecisions for ions in the test data set (Table ) are shown in the central column. In the left- and right-hand side columns, only the μ-prediction errors representing the specified ions are shown. The central column was generated using a much-higher data-point density than was observed in the test data set and represents a higher data-sampling rate, as may be observed in other instruments. It also equivalently represents a higher m/z ion or a lower mass-spectral resolution instrument. The upper row of plots in Fig.  includes a horizontal dashed line to illustrate the peak width (w in Eq. ). Also shown in this row are red vertical bars spanning ± 17ppm, which is the standard deviation of the aggregated error estimates for well-resolved peaks in the test data set (shown in the lower-right plot of Fig. ). The grey vertical bars span ± 10ppm to illustrate the range of biases observed in Table . (The position of a peak relative to the grey bars will, by definition, change only between ions and between data sets, not between samples.)

The second row of plots in Fig.  plots σh for each of the three cases. Because the peak is nearly Gaussian, the plot is virtually symmetrical about the zero-bias line. Nevertheless, we have plotted both negative and positive biases because this is the space in which the true, unknown biases of all peaks will lie.

The third row of plots in Fig.  shows the effect of μ-prediction biases on σh, relative to the no-bias case. That is, this row shows σh,with bias divided by σh,bias=0, where the former quantity is the σh obtained when considering μ-prediction biases and the latter quantity is the σh obtained when neglecting those biases. It is clear that the larger the m/z-calibration imprecision, the smaller the importance of the m/z-calibration bias.

Thus, the bias need not be known or estimated for the accurate Monte Carlo estimation of σh; it is only necessary that the m/z-calibration imprecision employed for the Monte Carlo estimate is large enough that σh lies within the regime where the influences of m/z-calibration biases are negligible. This will result in σh values that are overestimated by a few percent but that are consistent between peaks with different m/z-calibration bias. For the biases observed in Fig. , a reasonable estimate for an m/z-calibration imprecision in this regime may be 20ppm; an example of the resulting σh is shown in shown in Fig. , case E.

Consistent estimates of σh between peaks with different calibration biases is important during least-squares fitting. This statement may be justified by the conclusions of , in particular the conclusion that underestimating the imprecision of certain variables during PMF can have significantly harmful effects. Those authors also concluded that overestimating uncertainties may have potentially positive effects. Both conclusions provide clear support for the present method.

An approach to estimating σh which does not require ion-specific estimates of m/z-calibration biases or imprecisions is also important during fits to multiple overlapping peaks, since the estimation of m/z-calibration biases in such a context (Sect. ) may be difficult or impossible. This is further discussed in Sect. .

With σw/w determined as ∼2.5 % (Sect. ) and σh/h estimated by the Monte Carlo method described above (Sect. ), for both well-resolved and overlapping peaks, the overall imprecision in the peak integration discussed in Sect.  can be estimated from Eq. () as σAA2=σhh2+σww2, where σh/h is approximately constant (on the order of a few percent) for a well-resolved peak within a given data set and may become much larger as peak overlap becomes important .

σA is considered independent of the Poisson counting uncertainties σp discussed in Sects.  and , so that the overall HR-AMS uncertainty may be expressed in units of ion counts per detector bin as σAMS=σA2+σp2.

For well-resolved peaks, the first term in Eq. () scales linearly with signal, whereas the second term scales with the square root of the signal. Therefore, even for well-resolved peaks, the first term dominates at high signals. That is, in this case peak-integration imprecision dominates ion-counting imprecision. For overlapping peaks, peak-integration imprecision may become very large for much smaller signals.

While σh/h is most accurately estimated on a case-by-case basis using the Monte Carlo approach outlined above, we illustrate the importance of peak-integration errors by taking σh/h≈4% as a representative value for well-resolved peaks in the current data set (estimated based on Sect. ), which together with σw/w≈2.5% gives σA≈(4.7%)A for well-resolved peaks in this data set. This is shown in Fig. , which has been obtained by programming Eq. () into PIKA using the value for σA given above. (As is standard in PIKA, the calculations have accounted for additional details related to sampling time, baseline noise, and mass-spectral duty cycle . Due largely to sampling-time correction , the asymptote of Fig. b is less than 4.7 %.)

Fig. b illustrates the essential difference between fitting and counting errors. Whereas the relative uncertainty of Poisson errors (∼n) falls to 0 as the signal increases, the relative uncertainty of fitting errors (∼n) tends to an asymptotic value as signal increases. This feature strongly affects the relative importance of high-signal data during uncertainty-weighted fitting. This importance will be even greater for peaks affected by overlap, for which fitting errors may increase rapidly and nonlinearly.

The arguments presented above clearly show that m/z-calibration limitations lead to a linear (fractional) imprecision term during the constrained peak-integration procedures of PIKA. This imprecision significantly increases the overall imprecision in a resulting data matrix; in particular, fractional imprecisions become more significant than ion-counting uncertainties for high signals, and overlap errors may be larger still. Thus, omitting this imprecision term from the uncertainty provided to PMF leads to an overweighting of higher-signal data, which may bias the PMF solution .

The importance of high signals means that fitting errors may be especially important for the high aerosol concentrations that may be measured at the roadside e.g., within combustion plumes e.g., in highly polluted cities like Beijing or Mexico City e.g., or during source studies e.g..

Given the variable importance of high signals within different data sets, and the importance of instrument-specific parameters to the peak-integration imprecision as discussed above, an absolute statement of the relevance of these uncertainties in PMF is impossible. Nevertheless, as an example we performed PMF on a synthetic data matrix to demonstrate the significance of addressing peak-fitting errors.

The synthetic data matrix was constructed using the PMF solution presented by , which corresponds to the test data set discussed herein. In detail: the factors (mass spectra) and loadings (time series) of the PMF solution were recombined into an error-free synthetic data matrix. Simulated noise was added to this matrix to represent Poisson and peak-fitting imprecisions (given by Eq. ) by sampling from Poisson- and Gaussian-noise generators (Igor Pro, version 6.3) with an SD of 5 % of the peak height. For simplicity, peak-overlap errors were not included. PMF was performed as described in . In particular, low signal-to-noise variables were downweighted and the robust mode of PMF was used , consistent with standard practice in the AMS community .

When this synthetic matrix was factorized using only a Poisson imprecision term to weight the data (i.e. with the largest signals overweighted), the residual matrix showed significant outliers for the highest signals. For example, the largest residual outliers for CO+, which was the highest-signal ion in the synthetic matrix, corresponded to spikes in the signal of that ion. These residual outliers were not present when the synthetic matrix was factorized using the correct imprecision model, Eq. ().

The appearance of the highest CO+ signals as spikes in the residual matrix when using the incorrect imprecision model is highly significant. Under normal circumstances, the analyst may have regarded such spikes as “outliers” reflecting transient signals or data-analysis problems, even though (in this case) they were purely the result of underestimated uncertainties. Properly weighting these spikes caused the r2 between the output and input time series of the lowest-signal factor to increase from 0.54 to 0.74, with similar but smaller increases for the higher-signal factors, which were retrieved with r2>0.91 in the wrongly weighted case and with r2>0.999 in the correctly weighted case. Thus, the influence of high signals on the retrieval of low-signal PMF factors may thus directly influence the mass concentrations estimated by HR-AMS–PMF. This may significantly bias the mass concentrations of specific HR-AMS factors, particularly when they are retrieved in the presence of higher-signal factors. No exact number can be provided for this effect, as it will depend on the relative magnitude of the signals as well as the mass-spectral profiles of each of the retrieved components.

The largest uncertainty in this analysis was due to the estimation of the biases and imprecisions in μ prediction (which reflect errors in m/z calibration) directly from the data (Sect. ). We therefore emphasize that the ion-to-ion variability in μ-prediction biases which strongly influence σh are not unique to our study.

Biases in m/z calibration, also referred to as mass accuracy, are a general feature of time-of-flight mass spectrometers in atmospheric science and otherwise e.g.. This means that AMS-specific details such as particle vaporization prior to electron-impact ionization are not their sole cause. These biases may reflect a departure of the ion trajectories from their predicted paths but may also result when m/z calibration is performed on low-signal peaks . When the former mechanism is the cause, numerical studies may underestimate m/z-calibration uncertainties and thus underestimate σh.

introduced additional fit coefficients to the standard m/z-calibration equation and successfully reduced the magnitude of calibration biases. However, this approach is not feasible for all data sets, as it requires multiple peaks to be reliably available for calibration. Moreover, the biases remaining after applying such a modified calibration function would still need to be known for accurate Monte Carlo estimation of σh; reducing m/z-calibration bias does not remove the need to account for it. Nevertheless, an improved calibration would reduce the overall σh.

The above conclusions that the major causes of fitting errors in PIKA are errors in peak-location prediction raises the question of whether the fit procedure itself might be improved.

It would be preferable to allow a priori knowledge of calibration uncertainties to be incorporated into the fitting procedure, following Bayesian theory . In such an approach, realistic uncertainty distributions could be directly applied to the constraint of μ and translated directly into uncertainty distributions of the resulting h. However, this does not solve the problem of m/z-calibration biases being unknown for the case of overlapping peaks. The development of such an approach is beyond the scope of the present work, in part because the Igor Pro software in which PIKA has been developed over the past decade does not provide the necessary framework. We note that although the relaxation of the μ constraint in Sects.  and allowed for uncertainties in μ to be accounted for, the implicit probability distribution assigned to these uncertainties was an unphysical, uniform distribution with unrealistic, discontinuous edges. For the typical case where multiple overlapping peaks are fitted, this approach is not considered robust.

In addition to an improved fitting procedure, an improved calibration procedure would be an obvious recommendation for reducing μ-prediction errors. An accurate calibration requires both a consistent calibrant signal and a consistent peak shape. These requirements were not met by any of the mass-spectral peaks: the only consistent signals were those of background ions from the gas phase, which were relatively few and displayed significantly different peak shapes than particulate signals, likely owing to their originating from generally different regions of the AMS ionization chamber. An improved calibration might make use of an internal standard, for example a polyfluorinated organic , to be fitted by the same pseudo-Gaussian function used for data analysis (Eq. ). However, at some point the limited temporal resolution of the ion detector and the variability of ion flight paths also limit the accuracy of the m/z calibration.

By definition, the Monte Carlo approach invariably requires more time than a direct approach. In large HR-AMS data sets, thousands of mass spectra may be recorded, each containing hundreds of peaks to be fitted. On modern personal computers, fitting may consequently take hours, so performing 100 Monte Carlo simulations during exploratory data analysis can be impractical. A compromise between rapid and reliable results is therefore desirable.

In their work, developed a parameterization designed to estimate σh based on the separation of two peaks, for the reasons noted above. This parameterization does not account for the constant relative imprecision expected for well-resolved peaks (Sect. ). Additionally, the parameterization has been designed for the two-peak system.

To minimize the computation time required for the Monte Carlo approach, the following algorithmic approach could be used. First, the approximation σh/h≈ constant could be applied for all well-resolved peaks. A percentage value of σh/h could be estimated using a subset of the data set; weighted least-squares fits will typically be insensitive to the exact value. For cases where two peaks are close enough to possibly overlap, the parameterization of may be used to quantitatively define when their overlap is likely to influence σh (for example, when the parameterization indicates σh/h>5% for any two peaks). When overlap is significant, a Monte Carlo estimation of σh can be initiated. Further, once a Monte Carlo estimate is obtained for a given set of peak heights, the result may be stored and applied to subsequent peaks of similar height ratios; where more restrictive definitions of similarity will be more accurate but slower. The validity of these approximations should be directly evaluated after initial data exploration by performing full Monte Carlo calculations for each group of peaks in the data set.

While only peak-integration and ion-counting uncertainties were addressed in the discussion above, a number of other AMS-specific uncertainties can be identified.

discussed and addressed a number of HR-AMS uncertainties with regard to background signals and overlapping ions. also developed a data-smoothing approach to evaluate data quality for PMF, rejecting noisy variables. Such an approach may remain useful for the very-slowly changing systems studied by , as it may account for imprecisions other than those caused by peak integration; however, it did not result in improved PMF uncertainty estimates but rather the selective rejection of low signal-to-noise variables. More generally, smoothing approaches cannot be applied to the rapidly changing signals observed during roadside studies, aircraft studies, or laboratory studies on rapidly changing sources, all of which may entail the measurement of the high PM loadings for which peak-integration uncertainties are most important.

Other AMS-specific uncertainties include the fundamental uncertainty involved in converting electronic signals at the detector to ion counts. This conversion is performed after estimating the signal intensity of a single ion, a process complicated by the signal-thresholding applied by the data acquisition software (version 4.0.9). We performed this single-ion measurement procedure on each of the measurement days described herein and obtained results varying by ∼ 20 %. We emphasize that this 20 % would correspond to a potential bias in the integrated areas of all ions, and is fundamentally different to the peak-integration imprecisions discussed extensively above. Moreover, as it is not clear whether this number represents instrumental or procedural variability (the standard procedure resulted in different ions being used for single-ion measurement on each day), it has not been included in Eq. (). In addition, it is worth noting that the newest version of the HR-AMS includes significant hardware improvements which address this issue.

A second HR-AMS-specific scenario arises when the intensity of less-abundant isotope peaks are predicted based on the intensity of more-abundant isotope peaks. This procedure propagates fitting errors across integer m/z, and the corresponding uncertainty should also be propagated to these and neighbouring peaks, as discussed in .

Finally, not all ions follow the peak shape established by the PIKA calibration procedure. In particular, thermally generated ions such as K+ or other slowly evaporating species may lead to additional uncertainties for this reason.