Pitfalls in Nonlinear Inversion

— We discuss and illustrate graphically with simple 2-D problems, four common pitfalls in geophysical nonlinear inversion. The first one establishes that the Lagrange multiplier, used to incorporate a priori information in the geophysical inverse problem, should be the largest value still compatible with a reasonable data fitting. This procedure should be used only when the interpreter is sure about the importance of the a priori information used to stabilize the inverse problem relative to the geophysical observations. Because this is rarely the case, the user should use the smallest Lagrange multiplier still producing stable solutions. The second pitfall is an attempt to automatically estimate the Lagrange multiplier by decreasing it along the iterative process used to solve the nonlinear optimization problem. Consequently, at the last iteration, the Lagrange multiplier may be so small that the problem may become ill-posed and any computed solution in this case is meaningless. The third pitfall is related to the incorporation of a priori information by a technique known as “Jumping.” This formulation, from the viewpoint of the class of Acceptable Gradient Methods, is incomplete and may lead to a premature halt in the iteration, and, consequently, to solutions far from the true one. Finally, the fourth pitfall is an inadequate convergence criterion which stops the iteration when the data misfit drops just below the noise level, irrespective of the fact that the functional to be minimized may not have attained its minimum. This means that the a priori information has not been completely incorporated, so that this stopping criterion partially neutralizes the effect of the stabilizing functional, and opens the possibility of obtaining unstable, meaningless estimates.     

Variations in heavy metals concentrations in Antarctic snows from 1940 to 1980

Concentrations of Pb, Cd, Cu and Zn have been measured using improved ultraclean procedures in a succession of twenty six snow samples integrating a 40 yr time sequence from 1940 to 1980 which were collected from the walls of a 6 m deep pit at stake D 55 in Adelie Land, East Antarctica. Measured concentrations, which are among the lowest ones ever measured in Antarctic snows, are found not to have significantly increased during the investigated time period, with the possible exception of Pb for which there might have been a significant increase after the mid 1960's. For this last metal, measured concentrations in the 1940's are about 6-fold higher than in Antarctic Holocene ice several thousand years old, which indicates that a large fraction of the anthropogenic increase for Pb probably occurred before the 1940's.