GNGTS 2015 - Atti del 34° Convegno Nazionale

(3) and if we consider a vertical sounding with N measurements { P k1 ,…, P kj ,…, P kN }, then Eq. 1 becomes: (4) This equation describes the 1D forward problem of a continuous unknown function ρz linearly related to the gravity data. If we assume a discrete number M of layers, where in each of them the density is homogeneous, we have: (5) where: (6) and { ρ 1j ,…, ρ ij …, ρ Mj }, { V 1 ,…, V i ,..., V M } are respectively the densities and the volumes of the M layers. with i= [1 ,…,M ]. Fedi and Rapolla (1990) showed that, normalizing the field in Fig. 1a with that produced by the same volume with unit-density, the resulting quantities (apparent densities, Fig. 1b) are strongly related to the behavior of density, that is decreasing (left model, Fig. 1c) or increasing (right model, Fig. 1c). We now turn on the inversion of 1D gravity soundings. We here use the method, called “Minimum Length with Inequality Constrains” or IML. It is used to solve indeterminate linear problems such as: (7) where g represents the data vector referred to the j th sounding, ρ represents the density vector referred to the volume and G represents the matrix of the theoretical kernel, defined by the Eq. 6. This kind of estimation model is also known as “minimum length” of the solution for indeterminate problem. The presence of experimental errors ( ∆g ), implies that the fitting between experimental data and theoretical data ( G ρ ) is searched according to: (8) Besides, to reduce the number of possible models that are solution to our problem, constraints can be used to express our a priori information on the density of each layer: (9) With these inequalities, the problem is first transformed into a non-negative least squares problem , according to a technique described in Menke (1984): minimize || m || 2 subject to: Fm ≥ h This problem may be transformed into: (10) (11) And find the solution that minimizes: 168 GNGTS 2015 S essione 3.3