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Register a new userPros and cons of the technique of processing IRMS data as desired-delta values: uncertainty and comparability in results, a case study for determining carbon and oxygen isotopic abundance ratios as CO_2^+
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Authors
B. P. Datta
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In isotope ratio mass spectrometry (IRMS), any sample (S) measurement is performed as a relative-difference ((S/W)di) from a working-lab-reference (W), but the result is evaluated relative to a recommended-standard (D): (S/D)di. It is thus assumed that different source specific results ((S1/D)di, (S2/D)di) would represent their sources (S1, S2), and be accurately intercomparable. However, the assumption has never been checked. In this manuscript we carry out this task by considering a system as CO2+-IRMS. We present a model for a priori predicting output-uncertainty. Our study shows that scale-conversion, even with the aid of auxiliary-reference-standard(s) Ai(s), cannot make (S/D)di free from W; and the ((S/W)di,(A1/W)di,(A2/W)di) To (S/D)di conversion-formula normally used in the literature is invalid. Besides, the latter-relation has been worked out, which leads to e.g., fJ([(S/W)dJCO2pmp%],[(A1/W)dJCO2pmp%],[(A2/W)dJCO2pmp%]) = ((S/D)dJCO2pm4.5p%); whereas FJ([(S/W)dJCO2pmp%],[(A1/W)dJCO2pmp%]) = ((S/D)dJCO2pm1.2p%). That is, contrary to the general belief (Nature 1978, 271, 534), the scale-conversion by employing one than two Ai-standards should ensure (S/D)di to be more accurate. However, a more valuable finding is that the transformation of any d-estimate into its absolute value helps improve accuracy, or any reverse-process enhances uncertainty. Thus, equally accurate though the absolute-estimates of isotopic-CO2 and constituent-elemental-isotopic abundance-ratios could be, in contradistinction any differential-estimate is shown to be less accurate. Further, for S and D to be similar, any absolute estimate is shown to turn out nearly absolute accurate but any (S/D)d value as really absurd. That is, estimated source specific absolute values, rather than corresponding differential results, should really represent their sources, and/ or be closely intercomparable.
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The suitability of a mathematical-model Y = f({Xi}) in serving a purpose whatsoever (should be preset by the function f specific input-to-output variation-rates, i.e.) can be judged beforehand. We thus evaluate here the two apparently similar models: YA = fA(SRi,WRi) = (SRi/WRi) and: YD = fd(SRi,WRi) = ([SRi,WRi] - 1) = (YA - 1), with SRi and WRi representing certain measurable-variables (e.g. the sample S and the working-lab-reference W specific ith-isotopic-abundance-ratios, respectively, for a case as the isotope ratio mass spectrometry (IRMS)). The idea is to ascertain whether fD should represent a better model than fA, specifically, for the well-known IRMS evaluation. The study clarifies that fA and fD should really represent different model-families. For example, the possible variation, eA, of an absolute estimate as the yA (and/ or the risk of running a machine on the basis of the measurement-model fA) should be dictated by the possible Ri-measurement-variations (u_S and u_W) only: eA = (u_S + u_W); i.e., at worst: eA = 2ui. However, the variation, eD, of the corresponding differential (i.e. YD) estimate yd should largely be decided by SRi and WRi values: ed = 2(|m_i |x u_i) = (|m_i | x eA); with: mi = (SRi/[SRi - WRi]). Thus, any IRMS measurement (i.e. for which |SRi - WRi| is nearly zero is a requirement) should signify that |mi| tends to infinity. Clearly, yD should be less accurate than yA, and/ or even turn out to be highly erroneous (eD tends to infinity). Nevertheless, the evaluation as the absolute yA, and hence as the sample isotopic ratio Sri, is shown to be equivalent to our previously reported finding that the conversion of a D-estimate (here, yD) into Sri should help to improve the achievable output-accuracy and -comparability.
C and O isotopic ratios are reported for a sample of 46 Mira and SRa-type variable AGB stars. Vibration-rotation 1st and 2nd overtone CO lines in 1.5 to 2.5 $mu$m spectra were measured to derive isotopic ratios for 12C/13C, 16O/17O, and 16O/18O. Comparisons with previous measurements for individual stars and with various samples of evolved stars are discussed. Models for solar composition AGB stars of different initial masses are used to interpret our results. We find that the majority of the M stars had main sequence masses < 2 Msun and have not experienced sizable third dredge-up episodes. The progenitors of the four S-type stars in our sample are slightly more massive. Of the 6 C stars in the sample three have clear evidence relating their origin to the occurrence of the third dredge-up. Comparisons with O-rich presolar grains from AGB stars that lived before the formation of the solar system reveal variations in the interstellar medium chemical composition. The present generation of low-mass AGB stars show a large spread of 16O/17O ratios, similar to that of group 1 presolar grains and in agreement with theoretical expectations for the composition of mass 1.2 to 2 Msun stars after the 1st dredge up. On the contrary, the 16O/18O ratios of present-day LPVs are definitely smaller than those of group 1 grains. This occurrence is most probably a consequence of the the decrease with time of the 16O/18O ratio in the interstellar medium due to the chemical evolution of the Milky Way. One star in our sample has an O composition similar to that of group 2 presolar grains originating in an AGB star undergoing extramixing. This occurrence may indicate that the extramixing process is hampered at high metallicity or, equivalently, favored at low metallicity. Similar to O-rich grains no star in our sample shows evidence of HBB, expected for massive AGB stars.
Evaluation of a variable Yd from certain measured variable(s) Xi(s), by making use of their system-specific-relationship (SSR), is generally referred as the indirect measurement. Naturally the SSR may stand for a simple data-translation process in a given case, but a set of equations, or even a cascade of different such processes, in some other case. Further, though the measurements are a priori ensured to be accurate, there is no definite method for examining whether the result obtained at the end of an SSR, specifically a cascade of SSRs, is really representative as the measured Xi-values. Of Course, it was recently shown that the uncertainty (ed) in the estimate (yd) of a specified Yd is given by a specified linear combination of corresponding measurement-uncertainties (uis). Here, further insight into this principle is provided by its application to the cases represented by cascade-SSRs. It is exemplified how the different stage-wise uncertainties (Ied, IIed, ... ed), that is to say the requirements for the evaluation to be successful, could even a priori be predicted. The theoretical tools (SSRs) have resemblance with the real world measuring devices (MDs), and hence are referred as also the data transformation scales (DTSs). However, non-uniform behavior appears to be the feature of the DTSs rather than of the MDs.
Carbon and oxygen isotopic ratios are reported for a sample of 51 SRb- and Lb-type variable asymptotic giant branch stars. Vibration-rotation first- and second-overtone CO lines in 1.5-2.5 mum spectra were measured to derive isotopic ratios for 12C/13C, 16O/17O, and 16O/18O. Comparisons with previous measurements for individual stars and with various samples of evolved stars, as available in the extant literature, are discussed. Using the oxygen isotopic ratios, the masses of the SRb stars can be derived. Combining the masses with Gaia luminosities, the SRb stars are shown to be antecedents of the Mira variables. The limiting parameters where plane-parallel, hydrostatic equilibrium model atmospheres can be used for abundance analysis of M giants are explored.
For a known weak signal in additive white noise, the asymptotic performance of a locally optimum processor (LOP) is shown to be given by the Fisher information (FI) of a standardized even probability density function (PDF) of noise in three cases: (i) the maximum signal-to-noise ratio (SNR) gain for a periodic signal; (ii) the optimal asymptotic relative efficiency (ARE) for signal detection; (iii) the best cross-correlation gain (CG) for signal transmission. The minimal FI is unity, corresponding to a Gaussian PDF, whereas the FI is certainly larger than unity for any non-Gaussian PDFs. In the sense of a realizable LOP, it is found that the dichotomous noise PDF possesses an infinite FI for known weak signals perfectly processed by the corresponding LOP. The significance of FI lies in that it provides a upper bound for the performance of locally optimum processing.
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