Joint Compositional Calibration: An Example for U–Pb Geochronology

Abstract

This contribution explores several issues arising in the measurement of a (geo)chemical composition with Laser Ablation Inductively Coupled Plasma Mass Spectrometry (LA-ICP-MS), specially in the case that the quantities of interest are linear functions of (log)-ratios. These quantities are scale invariant, but in general cannot be estimated without taking into account possible additive noise effects of the instrumentation, incompatible with a purely compositional approach. The proposed ways to a solution heavily build upon the multi-Poisson distribution, highlighting the counting nature of the readings delivered by these instruments. The model can be fitted using a generalized linear model formalism, and it allows for a joint calibration of all components at once. Relevance of these considerations is shown with some simulation studies and in a real case of multi-isotopic geochronological analyses. Results suggest that the most critical aspect of this analytical technique is the assumption that the amount of ablated mass per second between samples of unknown and known compositions is similar (matrix matching): if this cannot be ensured, absolute estimations of the abundance of each of these isotopes fails, while their (log)ratios are perfectly estimable. This opens the door to using the model for a joint calibration by loosening the condition of matrix matching and using several standards of different composition.

Appendix A

1.1 A.1 Three Competing Geometries

In this paper, three of the compositional geometries gathered by van den Boogaart and Tolosana-Delgado [12] are used, two on vectors of positive amounts ($\mathbf {x}\in \mathbb {R}_+^D$), either based on an interval or on a ratio scale; and a relative ratio scale on compositions s.s. ($\mathbf {x}\in \mathcal {S}^D$, the simplex). This section just summarizes their geometries.

The interval scale on $\mathbb {R}_+^D$ is captured by a geometry inherited from embedding $\mathbb {R}_+^D$ on $\mathbb {R}^D$ with its Euclidean space operations, namely the classical vector sum $+$ and multiplication by scalars $\cdot$. The fundamental isometric operation is the identity, thus this geometry is optimally reproduced by linear models with an identity link function.

The ratio scale on $\mathbb {R}_+^D$ is captured by equipping this set with the Abelian group operation (amount)-perturbation $\oplus _+$ and (amount)-powering $\odot _+$, respectively, the component-wise product of the components of two vectors and the component-wise powering of the components of one vector to the same scalar [6]. The fundamental isometric operation is the component-wise log-transformation, thus this geometry is optimally reproduced by linear models with a logarithmic link function.

The relative scale on $\mathcal {S}^D$ is captured by equipping this set with the Abelian group operation perturbation $\oplus _+$ and powering $\odot _+$, respectively the closed component-wise product of the components of two vectors [1] and the component-wise powering of the components of one vector to the same scalar. The fundamental isometric operation is the centered log-ratio transformation, thus this geometry is optimally reproduced by linear models with a logratio link function.

In general terms, a raw scale should be preferred for one variable which absolute differences are meaningful; a ratio scale for variable which meaningful differences are relative; and a relative scale on the simplex should be the choice if several variables show a ratio scale at the same time and their sum is either an artifact or a meaningless constant. Nevertheless, sometimes the same variables can be studied in one way or another, depending on the question that should be answered, i.e. the scale should be chosen depending on the question and not only on the data.

1.2 A.2 Alternative Joint Models

1.2.1 A.2.1 Models Ignoring the Background or With Multiplicative Effects

From the point of view of the relative scale on $\mathcal {S}^D$, all models presented in this paper are particularly complicated by the presence of the additive background. If this could be ignored (e.g. because it is very small with regard to the signal), and the sum of the components of $\mathbf {Z}$ equals one (i.e. a whole composition is analysed), then the following models are derived:

• scalar upscaling: $[\mathbf {X}(t_k)|n]\sim \mathcal {M}u(\mathbf {Z}; n)$ and $n\sim \mathcal {P}o(\omega _0\lambda (t_k))$;

• perturbation: $[\mathbf {X}(t_k)|n]\sim \mathcal {M}u(\mathbf {Z}\oplus \varvec{\lambda }(t_k); n)$ and $n\sim \mathcal {P}o(\omega _0\sum _i^D \lambda _i(t_k))$,;

• interaction-perturbation: $[\mathbf {X}(t_k)|n]\sim \mathcal {M}u(\mathcal {C}\left[ \varvec{\Lambda }^*\cdot (\mathbf {Z}\oplus \varvec{\lambda }(t_k))\right] ; n)$ and $n\sim \mathcal {P}o(\omega _0 \mathbf {1}^{\prime }\cdot (\varvec{\Lambda }^*\cdot (\mathbf {Z}\oplus _+ \varvec{\lambda }(t_k))))$.

This last model is still a mixture of additive and multiplicative geometries. The following models are purely compositional alternatives, using in the compositional part only operations on the simplex

• scalar upscaling: $[\mathbf {X}(t_k)|n]\sim \mathcal {M}u(\mathbf {Z}\oplus \varvec{\lambda }_b; n)$ and $n\sim \mathcal {P}o(\omega _0\lambda (t_k)\sum _i^D \lambda _{bi})$;

• perturbation: $[\mathbf {X}(t_k)|n]\sim \mathcal {M}u(\mathbf {Z}\oplus \varvec{\lambda }(t_k)\oplus \varvec{\lambda }_b; n)$ and $n\sim \mathcal {P}o(\omega _0\sum _i^D \lambda _{bi}\lambda _i(t_k))$;

• interaction-perturbation: $[\mathbf {X}(t_k)|n]\sim \mathcal {M}u(\varvec{\Lambda }^*\boxdot (\mathbf {Z}\oplus \varvec{\lambda }(t_k)) \oplus \varvec{\lambda }_b; n)$ and $n\sim \mathcal {P}o(\omega _0 \mathbf {1}^{\prime }\cdot (\varvec{\Lambda }^*\boxdot (\mathbf {Z}\oplus \varvec{\lambda }(t_k))\oplus \varvec{\lambda }_b))$, in this case with $\oplus$ the perturbation on the simplex.

In these expressions, we have used the notation $\boxdot$ after Pawlowsky-Glahn, Egozcue and Tolosana-Delgado ([7], Chap. 4) to denote a simplicial endomorphism operation, i.e. one such that once expressed in any basis of the simplex becomes a simple matrix-vector product. Note that these purely compositional models imply, among other effects, that the noise induced by the background upscales with the signal, i.e. larger signal should show more background variability.

In any of the cases presented, we finally have distributions for the number of counts on each element class that belong to the multi-Poisson family, with an intensity vector model $\omega _0\varvec{\Lambda }(\mathbf {Z}, \varvec{\lambda }_b, \varvec{\theta }; t_k)$ capturing the relationship between the expected partial counts and the composition of the analyte. Hence, we can always consider that the total number of counts $n(t_k)$ follows a Poisson distribution with $\lambda ^T_k=\omega _0 \mathbf {1}^{\prime }\cdot \varvec{\Lambda }(\mathbf {Z}, \varvec{\lambda }_b, \varvec{\theta }; t_k)$; and, conditional on that total, the vector of counts for each element follows a multinomial distribution with probability parameter vector $\mathbf {p}_k = \mathcal {C}\left[ \varvec{\Lambda }(\mathbf {Z}, \varvec{\lambda }_b,\varvec{\theta }; t_k)\right]$.

A relevant minor modification for the examples presented in this paper (Sects. 5 and 6) consists of the case that the dwell times are not equal for all isotopes. If we denote the vector of dwell times as $\varvec{\omega }_0 \in \mathbb {R}_+^D$, then it is immediate to show that the number of counts on each element class belong to the multi-Poisson family, with a perturbed intensity vector model $\varvec{\omega }_0\oplus _+\varvec{\Lambda }(\mathbf {Z}, \varvec{\lambda }_b, \varvec{\theta }; t_k)$, with perturbation on $\mathbb {R}_+^D$. Again, the total number of counts $n(t_k)$ will follow a Poisson distribution, albeit with total expected counts $\lambda ^T_k=\varvec{\omega }_0^{\prime }\cdot \varvec{\Lambda }(\mathbf {Z}, \varvec{\lambda }_b, \varvec{\theta }; t_k)$; and the vector of counts for each element conditionally follows a multinomial distribution with probability parameter vector $\mathbf {p}_k = \mathcal {C}\left[ \varvec{\omega }_0\oplus _+\varvec{\Lambda }(\mathbf {Z}, \varvec{\lambda }_b,\varvec{\theta }; t_k)\right]$.

1.2.2 A.2.2 Implications for the Estimation of GLMs

The methods presented in the main part of this contribution were characterized by an identity link function, a requirement of the additive nature of the background and the signal. If this condition is removed, then the class of models can be extended to models with logarithmic link function (actually, the canonical choice of Poisson GLMs). This setting would be suitable to consider the multiplicative models mentioned in Sect. A.2.1. The same structure as in Sect. 4 would then be used, namely a calibration phase in which all parameters would be estimated with a GLM; and a prediction phase in which another GLM with an offset (equal to the multiplicative background) would be used to estimate the unknown composition of the samples.