diego domenzain. September 2020 @ Colorado School of Mines
Gravity is dependent on density. By measuring the gravity field, it is possible to solve for (invert) density.
This script is an example of density inversion at depth, jointly using gravity gradient data on x and z.
the gravity field u = (ux,uy,uz) at point ro is given by,
u = G ∫_V ( ρ(r) * (r-ro) / ||r-ro||^3 )
G is the gravitational constant
where V is the integration volume
r is the integration variable
Assume the data u has been normalized by G.
At receiver locations the data is given by,
d_x = M*Lx*rho
d_z = M*Lz*rho
M is the measuring operator
Lx and Lz are matrices depending only on geometry
rho is the density in vector form (rho=rho(:))
The data sensitivity with respect to the density (jacobian) is,
Jx = M*Lx
and the gradient is
g_x = Jxt*e_x
where e_x is the residual ( e_x = d_x - d_x_observed ). t is for transposed.- 3-D inversion of gravity data. Yaoguo Li, and Douglas W. Oldenburg. Geophysics 1998.
The inversion itself is different than that of Yaoguo's,
- We do not use "depth weighting" because it is not necessary.
- We explicitly write the gradients.
Depth resolution is an issue nonetheless.
Below is an example of the true and recovered density. The colormap is the same for both pictures.
