[ ] bullet point about sparseness on page 3: I would change the last sentence to "we expect only a few, isolated lines to participate in (say) a given component of the surface-velocity-related distortion" because we definitely wouldn't expect ALL basis functions for surface-velocity effects to be isolated to a few wavelengths - something like convective blueshift will distort/change all the line profiles the same way to first order.
[ ] L and T are not defined for the summations in equations 5, 6, 9 (on purpose? how do we define those anyway - arbitrary choice?)
[ ] b_l not defined in equation 6
[ ] for "systematics" (equation 9), are those meant to be instrumental or continuum normalization-related in origin? If so, shouldn't it be a function of plain wavelength, not stellar rest-frame wavelength?
[ ] I'd also like to propose that the logo of Avast be a cartoon star wearing an eyepatch. (get it? because it's like a pirate but also like a starspot??)