## 2 Materials and Methods
### Elemental Spectra Processing and Mg/Ca Ratio (mmol/mol) Calculation  
In the recorded spectra, due to the high resolution and wide spectral range of the eight-channel spectrometer, many emission lines can be seen. Most of them are CaI and CaII lines, with several MgI and MgII lines and a few from SrI emission, in agreement with the expected chemical composition of mollusk shell carbonate. Although the presence of lithium in the carbonate of other species has been reported, no Li lines were found in our measurements. Due to the absence of specific matrix-matched calibration standards for marine mollusk shell biogenic carbonates, previous investigations carried out on marine mollusk shell biogenic carbonates obtained the Mg/Ca values from the raw ratio of two elemental lines, expressed in arbitrary units [57, 58, 68, 69]. In this study, a calibration-free LIBS (CF-LIBS) technique was applied for the first time on biogenic carbonates in order to estimate molar concentrations of magnesium and calcium chemical elements. The algorithm used is an implementation of the method originally proposed by [PERSON] et al. [71]. It obtains the concentrations from the Boltzmann and Saha-Boltzmann plots of many emission lines of the species of interest (in our case, CaI, CaII, MgI, and MgII) (Figure 4), and takes into account the electronic temperature and electron density of the plasma, thus reducing the variability of the signal due to shot-to-shot plasma fluctuations. To improve the accuracy of the results, the emission lines used by the CF algorithm have been carefully selected based on their signal to noise ratio in the spectra, energy levels, and self-absorption properties. One common issue of the CF-LIBS approach, in applications that use many measurements at different spatial points to create a sequence of concentrations, is that decisions to consider or discard specific emission lines are made individually for each spectrum. These decisions are dynamically made based on the particular signal to noise ratio of each emission peak, the quality of fit to a given peak function, or whether peak parameters are outliers in the Saha-Boltzmann plots or in the calculation of electron density. To reduce the variability of the concentrations determined across all the measurements of the sequences due to changes in the used emission lines, only the set of lines that were valid for all spatial points were considered here. This results in 16 lines of CaI, 5 of CaII and 6 of MgI being considered for the Saha-Boltzmann plots. As all MgII lines were discarded due to their high self-absorption, the concentration of ionized species was extrapolated using the Saha equilibrium equation [71]. The Matlab implementation of the CF-LIBS algorithm used in this work is openly available at github.com/acobo/CFLIBS. Figure 3: **(a)** Modern specimen of _Patella depressa_. Dashed lines indicate the cutting axis from which two thick sections were obtained. Dashed arrows show the two thick sections previously used for conducting a sclerochronological study on these shells. The remaining right-hand limpet section was used for elemental analyses carried out using LIBS. **(b)** Limpet cross-section showing the calcite layer and portion of the cross-section showing the visible growth lines. Red points indicate the LIBS measuring path. This procedure allowed us to obtain the relative concentration in mmol/mol units and provided significant advantages to the line ratio approach, decreasing LIBS-derived variability caused by diverse experimental conditions (room temperature, optical alignment, or spectrometer configuration) between specimens, but also the variability found along a single cross-section of a limet. In addition, to better assess the long-term (temperature related) ratio variations, the well-known singular spectrum analysis (SSA) was applied to the sequences [77; 78], a methodology that has already been applied on LIBS-derived series from marine mollusk shells elsewhere [57].