An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

summary:We consider the weighted space $W_1^{(2)}(\mathbb R,q)$ of Sobolev type $$ W_1^{(2)}(\mathbb R,q)=\left \{y\in A_{\rm loc}^{(1)}(\mathbb R)\colon \|y''\|_{L_1(\mathbb R)}+\|qy\|_{L_1(\mathbb R)}0\colon \inf _{x\in \mathbb R}\int _{x-a}^{x+a} q(t) {\rm d} t>0.$

Improved spatial resolution of elemental maps through inversion of LA-ICP-MS data

Laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS) provides the spatial distribution of elements within crystals and therefore can constrain the rates of geological processes. Spatial resolution of LA-ICP-MS is limited by the requirement to ablate sufficient material to surpass the detection limit of the instrument: too little material and the concentration cannot be measured; too much material from the same spatial location and the possibility of depth dependent variations in concentration increases. Because of this requirement and typical analytical setup, this commonly places a lower bound on the diameter of an ablation ‘spot’ size of approximately 20 μm for elements with ppm concentration. Here we present a means to achieve sub-spot size resolution using inverse methods. We discretize the space sampled in an analysis into pixels and note that the average concentration of the pixels sampled by a spot equals the measured concentration. As multiple overlapping spots sample some of the same pixels, we can combine discrete expressions for each spot as a system of linear equations. Through linear inversion with smoothness constraints we can solve for unknown pixel concentrations. We highlight this approach with two natural examples in which diffusive processes are important: magmatic ascent speeds and (U-Th)/He noble gas thermochronometry. In these examples, accurate results require that the true concentration gradients can be recovered from LA-ICP-MS data. We show that the ability to infer rapid rates of magma ascent is improved from months to weeks and that we are able to interpret previously un-interpretable thermochronometric data