Application of the two-step method for the solution of the inverse gravity problem for the Kolárovo anomaly

Density and gravity field Density Gravity field Input: Output:

First step Characteristic density (of the given surface gravity field) – properties: 1. It is the maximally smooth density generating the given surface gravity field: for the smallest possible 2. It is a linear integral transformation of the surface gravity field: 3. For the gravity field of a point source, it has its main extremum at the point source.

Solution Characteristic density or shortly chi-density: ( chi-density is a tetraharmonic function)

Second step The solution is represented by some number of anomalous bodies embedded in a horizontally layered medium; in the interior of any body and any layer, the density is a constant. The density in the formula for the gravity field is the difference density (it is equal to the density of the body minus the density of particular layer). The condition for finding the shape of any body is that the chi-density corresponding to the gravity field of the body has to be as near as possible to the original chi-density in the interior of the body. It has been found that it is sufficient that this condition holds at the boundary of the body. For any single body, its boundary consists of the upper and lower boundary. The problem is to find these boundaries for the given density within the body.

The problem is solved by the following iterative procedure: It is chosen some starting model of the body (values of the depth of the upper and lower boundary). For any model, it is calculated first its gravity field at the surface and the corresponding chi-density. Then it is compared the model chi-density with the original chi-density at the points of the upper and lower boundary. In the case of the body with positive difference density: for the upper boundary, if, upper boundary is shifted upwards; if, upper boundary is shifted downwards; for the lower boundary, if, lower boundary is shifted downwards; if, lower boundary is shifted upwards. If the shifts of the boundaries are sufficiently small, iteration is finished.

Gravity field Distances in km, gravity field in mgal

Chi-density z = -3 km Distances in km, density in kg/m^3

Chi-density z = -5 km Distances in km, density in kg/m^3

Chi-density z = -7 km Distances in km, density in kg/m^3

Chi-density z = -9 km Distances in km, density in kg/m^3

Chi-density x = -9 km Distances in km, density in kg/m^3

Chi-density x = -7 km Distances in km, density in kg/m^3

Chi-density x = -5 km Distances in km, density in kg/m^3

Chi-density x = -3 km Distances in km, density in kg/m^3

Chi-density x = -1 km Distances in km, density in kg/m^3

Chi-density x = 1 km Distances in km, density in kg/m^3

Chi-density x = 3 km Distances in km, density in kg/m^3

Chi-density x = 5 km Distances in km, density in kg/m^3

Chi-density x = 7 km Distances in km, density in kg/m^3

Chi-density x = 9 km Distances in km, density in kg/m^3

Chi-density x = 11 km Distances in km, density in kg/m^3

Chi-density y = -7 km Distances in km, density in kg/m^3

Chi-density y = -5 km Distances in km, density in kg/m^3

Chi-density y = -3 km Distances in km, density in kg/m^3

Chi-density y = -1 km Distances in km, density in kg/m^3

Chi-density y = 1 km Distances in km, density in kg/m^3

Upper boundary Distances in km

Upper boundary Distances in km

Upper boundary Distances in km

Upper boundary Distances in km

Body z = -3 km Distances in km, density in kg/m^3

Body z = -5 km Distances in km, density in kg/m^3

Body z = -7 km Distances in km, density in kg/m^3

Body z = -9 km Distances in km, density in kg/m^3

Body x = -9 km Distances in km, density in kg/m^3

Body x = -7 km Distances in km, density in kg/m^3

Body x = -5 km Distances in km, density in kg/m^3

Body x = -3 km Distances in km, density in kg/m^3

Body x = -1 km Distances in km, density in kg/m^3

Body x = 1 km Distances in km, density in kg/m^3

Body x = 3 km Distances in km, density in kg/m^3

Body x = 5 km Distances in km, density in kg/m^3

Body x = 7 km Distances in km, density in kg/m^3

Body x = 9 km Distances in km, density in kg/m^3

Body x = 11 km Distances in km, density in kg/m^3

Body y = -7 km Distances in km, density in kg/m^3

Body y = -5 km Distances in km, density in kg/m^3

Body y = -3 km Distances in km, density in kg/m^3

Body y = -1 km Distances in km, density in kg/m^3

Body y = 1 km Distances in km, density in kg/m^3

Body z = -3 km Distances in km, density in kg/m^3

Body z = -5 km Distances in km, density in kg/m^3

Body z = -7 km Distances in km, density in kg/m^3

Body z = -9 km Distances in km, density in kg/m^3

Body x = -9 km Distances in km, density in kg/m^3

Body x = -7 km Distances in km, density in kg/m^3

Body x = -5 km Distances in km, density in kg/m^3

Body x = -3 km Distances in km, density in kg/m^3

Body x = -1 km Distances in km, density in kg/m^3

Body x = 1 km Distances in km, density in kg/m^3

Body x = 3 km Distances in km, density in kg/m^3

Body x = 5 km Distances in km, density in kg/m^3

Body x = 7 km Distances in km, density in kg/m^3

Body x = 9 km Distances in km, density in kg/m^3

Body x = 11 km Distances in km, density in kg/m^3

Body y = -7 km Distances in km, density in kg/m^3

Body y = -5 km Distances in km, density in kg/m^3

Body y = -3 km Distances in km, density in kg/m^3

Body y = -1 km Distances in km, density in kg/m^3

Body y = 1 km Distances in km, density in kg/m^3

Body z = -3 km Distances in km, density in kg/m^3

Body z = -5 km Distances in km, density in kg/m^3

Body z = -7 km Distances in km, density in kg/m^3

Body z = -9 km Distances in km, density in kg/m^3

Body x = -9 km Distances in km, density in kg/m^3

Body x = -7 km Distances in km, density in kg/m^3

Body x = -5 km Distances in km, density in kg/m^3

Body x = -3 km Distances in km, density in kg/m^3

Body x = -1 km Distances in km, density in kg/m^3

Body x = 1 km Distances in km, density in kg/m^3

Body x = 3 km Distances in km, density in kg/m^3

Body x = 5 km Distances in km, density in kg/m^3

Body x = 7 km Distances in km, density in kg/m^3

Body x = 9 km Distances in km, density in kg/m^3

Body x = 11 km Distances in km, density in kg/m^3

Body y = -7 km Distances in km, density in kg/m^3

Body y = -5 km Distances in km, density in kg/m^3

Body y = -3 km Distances in km, density in kg/m^3

Body y = -1 km Distances in km, density in kg/m^3

Body y = 1 km Distances in km, density in kg/m^3

Body z = -3 km Distances in km, density in kg/m^3

Body z = -5 km Distances in km, density in kg/m^3

Body z = -7 km Distances in km, density in kg/m^3

Body z = -9 km Distances in km, density in kg/m^3

Body x = -9 km Distances in km, density in kg/m^3

Body x = -7 km Distances in km, density in kg/m^3

Body x = -5 km Distances in km, density in kg/m^3

Body x = -3 km Distances in km, density in kg/m^3

Body x = -1 km Distances in km, density in kg/m^3

Body x = 1 km Distances in km, density in kg/m^3

Body x = 3 km Distances in km, density in kg/m^3

Body x = 5 km Distances in km, density in kg/m^3

Body x = 7 km Distances in km, density in kg/m^3

Body x = 9 km Distances in km, density in kg/m^3

Body x = 11 km Distances in km, density in kg/m^3

Body y = -7 km Distances in km, density in kg/m^3

Body y = -5 km Distances in km, density in kg/m^3

Body y = -3 km Distances in km, density in kg/m^3

Body y = -1 km Distances in km, density in kg/m^3

Body y = 1 km Distances in km, density in kg/m^3