Efficient Geophysical Technique of Vertical Line Elements as a Natural Consequence of General Constraints Techniques

One of the main objectives of geophysics is to find how density ρ and other physical characteristics depend on a 3-D location (x, y, z). In general, in numerical methods, a way to find the dependence ρ(x, y, z) is to discretize the space, and to consider, as unknown, e.g., values ρ(x, y, z) on a 3-D rectangular grid. In this case, the desired density distribution is represented as a combination of point-wise density distributions. In geophysics, it turns out that a more efficient way to find the desired distribution is to represent it as a combination of thin vertical line elements that start at some depth and go indefinitely down. In this paper, we show that the empirical success of such vertical line element techniques can be naturally explained if we recall that, in addition to the equations which relate the observations and the unknown density, we also take into account geophysics-motivated constraints. c ⃝2012 World Academic Press, UK. All rights reserved.