SLONIP

SLOvenian Network of Isotopes in Precipitation

Supplementary data use info:

All meteorological data on SLONIP Website, either presented in graphical form or used for calculations, were obtained from the Slovenian Environmental Agency (ARSO) internet database available at the meteo.arso.gov.si/.
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All of the numeric data seen on this website is displayed and calculated dynamically using default, pre-written formulas. Although the statistics procedures should be universally unique, we are aware of the fact that different laboratories often use slightly different statistical formulas and methods, and because they can lead to misinterpretation of the results, they should not be neglected. For this purpose we decided to explicitly write down all of the formulas used to generate the results displayed for each observation station.
The main focus for the one who decides to study and interpret our results should be firstly on how we filtered the measured data i.e. how we decided which of the measured data to take in account when calculating averages and regression coefficients. Secondly, a great attention should be devoted to mathematical formulas used to calculate the latter.

Filters

Two main filters have been applied on the data for the purpose of correct analysis. Firstly, for each data type, when calculating means and regression coefficients we only took in account years in which $8$ or more monthly samples have been measured. And secondly, for each data type, the sum of precipitation for all of the months with known value of that data type must exceed $70$% of the total precipitation sum for that year.

Means

Means and weighted means are calculated in the usual way though we are still going to write down the formulas to avoid any misinterpretation of the results. For data type $x$:

$\bar{x}_{unweighted} = \frac{\sum_{i=1}^n x_i}{n}\\ \bar{x}_{weighted} = \sum_{i=1}^n x_i w_i \quad \text{where} \quad w_i = \frac{p_i}{\sum_{i=1}^n p_i} \quad , \quad p_i \ \text{is the total precipitation amount in that month.}$

One should also be aware of the definition of the meteorological seasons, which we used when calculating seasonal means (Winter: Dec-Feb, Spring: Mar-May, Summer: Jun-Aug, Autumn: Sept-Nov).

Deuterium excess

Deuterium excess is calculated in the standard manner: $\ d\text{-excess}= \delta^2$H$- 8\ \delta^{18}$O.

Regression methods

We have decided to use two different regression methods: the major axis regression (MA) and the reduced major axis regression (RMA). For both we have calculated the unweighted and the weighted regression. All of the formulas are taken from the article [ "Alternative least squares methods for determining the meteoric water line, demonstrating GNIP data", authors: Jagoda Crawford, Catherine E. Hughes, Spyros Lykoudis, published in Jurnal of Hydrology, 2014 ].
In the formulas stated below $a$ denotes the slope coefficient of the line $b$ denotes the intercept coefficient and $SE_a$ and $SE_b$ stand for their standard errors. The meteoric water line is of course of the form $\ y = a x + b \ ;\ y \text{~}\delta^2$H$\ , \ x \text{~} \delta^{18}$O.

Unweighted regression:

Reduced major axis regression (RMA): $\quad a_{RMA} = \sqrt{\frac{\sum_{i=1}^n y_i^2-\frac{(\sum_{i=1}^n y_i)^2}{n}}{\sum_{i=1}^n x_i^2-\frac{(\sum_{i=1}^n x_i)^2}{n}}}.$

Major axis regression (MA): $\quad a_{MA} = \frac{(\sum_{i=1}^n V_i^2-\sum_{i=1}^n U_i^2)+\sqrt{(\sum_{i=1}^n V_i^2-\sum_{i=1}^n U_i^2)^2+4(\sum_{i=1}^n U_i V_i)^2}}{2\sum_{i=1}^n U_i V_i} \quad \text{where} \quad U_i = (x_i-\bar{x}) \quad \text{and} \quad V_i = (y_i-\bar{y}).$

In both cases the intercept coefficient $b$ is given by: $\quad b = \frac{\sum_{i=1}^n y_i}{n} - a \frac{\sum_{i=1}^n x_i}{n} = \bar{y} - a \bar{x}.$

Standard errors were obtained from formulas: $\quad SE_a = \sqrt{\frac{\frac{\sum_{i=1}^n(y_i-\hat{y}_i)^2}{n-2}}{\sum_{i=1}^n(x_i-\bar{x})^2}} \quad, \quad SE_b = SE_a \sqrt{{\sum_{i=1}^nx_i^2}{n}} \quad \text{where} \quad \hat{y_i} = a x_i + b.$

Weighted regression:

As above in means we define the weight for each measurment : $\quad w_i = \frac{p_i}{\sum_{i=1}^np_i} \quad , \quad p_i$ is the corresponding precipitation amount.

It is also convenient do define: $\quad U_{wi} = x_i-\bar{x}_w \quad , \quad V_{wi} = y_i-\bar{y}_w \quad , \quad \text{where} \quad \bar{x}_w=\sum_{i=1}^n w_ix_i \quad \text{and} \quad \bar{y}_w=\sum_{i=1}^n w_iy_i.$

Precipitation weighted reduced major axis regression (PWRMA): $\quad a_{PWRMA} = \sqrt{\frac{\sum_{i=1}^n w_i(y_i-\bar{y}_w)^2}{\sum_{i=1}^n w_i(x_i-\bar{x}_w)^2}} = \sqrt{\frac{\sum_{i=1}^nw_iV_{wi}^2}{\sum_{i=1}^nw_iU_{wi}^2}}.$

Major axis regression (MA): $\quad a_{PWMA} = \frac{\sum_{i=1}^nw_iV_{wi}^2-\sum_{i=1}^nw_iU_{wi}^2+\sqrt{(\sum_{i=1}^nw_iV_{wi}^2-\sum_{i=1}^nw_iU_{wi}^2)^2+4(\sum_{i=1}^nw_iU_{wi}V_{wi})^2}}{2\sum_{i=1}^nw_iU_{wi}V_{wi}}.$

Intercept coefficient $b$ in both PW cases above is given by: $\quad b_{PW} = \sum_{i=1}^nw_iy_i - a \sum_{i=1}^nw_ix_i = \bar{y}_w - a \bar{x}_w.$

Standard errors: $\quad SE_{a,PW} = \sqrt{\frac{\frac{n\sum_{i=1}^nw_i(y_i-\hat{y}_i)^2}{n-2}}{n\sum_{i=1}^nw_i(x_i-\bar{x})^2}} \quad, \quad SE_{b,PW} = SE_{a,PW} \sqrt{\sum_{i=1}^nw_ix_i^2} \quad , \quad \hat{y_i} = ax_i + b.$