I think this has been discussed with Joe before, but do you have a preference for calculating the error correlation from observation rather than from differentiation? What do others think? is there some general consensus on which one is better? A few things that I have observed: 1) in synthetic datasets I can see absolutely no difference between the two methods. The calculated rho is always identical in the third or fourth decimal, so no preference there. 2) Pearson already observed in 1897 that r is around 0.5 in a ratio plot of uncorrelated variables, and TRUE correlation would have to be tested against the null hypothesis not being r=0, but against the spurious part (0.5 at minimum). This gets extreme when your denominator is 10x smaller than the numerator: the r calculated from sets of random (!) variables is on the order of 0.99. You would see very long slim error ellipses, and they are almost completely spurious.
This problem gets even more complex when you think about what you are actually measuring during laser ablation. Schmitz' 2007 assessment on ID TIMS UPb solution work assumes that you are measuring a solution, i.e., the same sample over and over. You also assume that the error on those multiple measurements would be normally distributed (or log-normally at least). This is not the case when we're ablating - the sample volume is strictly different every pulse, and we're not necessarily repeatedly sampling a homogeneous reservoir. This means that we should not expect normallly distributed errors. I guess the term 'error' is misleading, because the only error that is random comes from poisson statistics at the detector. In the best case when we're ablating down we could actually get a true single mineral isochron, in the worst case we're sampling random zonation. This is then mixed with counting errors on the detector, and a tiny little bit of 'true' error correlation. I think given all that non-randomness we should abandon the whole concept of error correlations in laser ablation and calculate conventional isochrons not from the means of the error ellipses but using all of the raw data points.
Or, and that is probably my preferred solution, switch entirely to inverse isochron techniques where we do not have a common denominator. But this has also been suggested for more than 40 years, and didn't stick. Let me know what you all think about this.