mmt_multipole_inversion.susceptibility_modules.spherical_harmonics_basis_volume

Module Contents

Functions

multipole_Bz_sus(dip_r, pos_r, Q, n_col_stride, dx_sensor, dy_sensor, dz_sensor, multipole_order)

Populate the susceptibility matrix using sensors with 3D-cuboid geometry

mmt_multipole_inversion.susceptibility_modules.spherical_harmonics_basis_volume.multipole_Bz_sus(dip_r, pos_r, Q, n_col_stride, dx_sensor, dy_sensor, dz_sensor, multipole_order)

Populate the susceptibility matrix using sensors with 3D-cuboid geometry

The sensors from the scanning surface are modelled as cuboids with volume. The susceptibility matrix is computed by integrating the Bz field in this volume, thus the flux is computed in units of Tesla m^3. In the main MultipoleInversion class the volume flux might be scaled by the volume to obtain the average volume within the scan sensor. The volume flux is computed as:

              _           _  t(1)  1(1)          t(1)  3(1)    t(2)  1(2)       _
Vol_flux =   /  dx dy dz |  Θ     P     + ... + Θ     P     + Θ     P     + ...  |
           _/            |_  1     z             3     z       1     z          _|

                             Dipole                            Quadrupole

where Θ are the multipole moments (in tensor basis) and Pz are the z-component spherical harmonic polynomials which have to be integrated. For instance:

                 _   _      _
 2(2)        ∂  |  \/2  x z  |
P      =  -  -- |  --------  |  ,  R = ( x^2 + y^2 + z^2 )^(1/2)
 z           ∂z |_    R^5   _|

Notice that the dz integration can be “cancelled” with the ∂z field derivative. The susceptibility matrix, therefore, will contain these polynomial integrated in the cuboid volume of the sensor, which are multiplied by the components of the magnetic moments column vector. Every row of the susceptibility matrix will contain the integrals for all magnetic sources and for one sensor position, since r = (x, y, z) = sensor_pos - magnetic_source_pos.

Parameters

dip_r

N x 3 array with the positions of the magnetic point sources

pos_r

P x 3 array with the positions of P sensors that define the scanning surface

Q

Susceptibility / Forward matrix to be populated. Check that Q entries are zero before calling this function

n_col_stride

Number of column strides to populate the Q matrix. This is defined by the multipole order of the potential expansion

dx_sensor, dy_sensor, dz_sensor

Half lengths of the sensor volume

multipole_order

Expansion order of the magnetic potential

Notes

The integration of the polynomials were obtained mostly with W. Mathematica