Effect of noise on determining firn structure and basal reflector depth

Here we calculate how much noise effects the calculation of firn structure, and hence the estimation of basal reflector depth. The basic idea is to start with a velocity model (the usual vostok one underlain by rock) and compute the reflection response at normal incidence. We then add in-band noise to the response, and then try to estimate the firn structure. We don't do a complete inversion, but rather limit the perturbation of firn structure to a single parameter, dv*exp(-z/30). We solve for the best-fitting dv (the one that minimizes the r.m.s. error between the reflection responses), and then convert the flucuation basal reflector traveltime to an error in its apparent depth. Here are the results for two different signal to noise levels:

Histograms of error in basal reflector depth, for two different r.m.s noise levels, 0.001 (solid) and 0.01 (bold). The reflection response has a maximum amplitude of about 0.1, so these correspond to signal-to-noise ratios of about 100:1 and 10:1, respectively. A signal-to-noise of 10:1 corresponds to a positioning accuracy of a few (<5) cm. (Postscript version).