Let

**d**^{o} be an N-dimensional vector whose

ith element is the vertical component of the magnetic field produced by a magnetic source in the position (

x^{i};

y^{i};

z^{i}) (

Figure 10a). Considering that the sample can be approximated by a set of L polygonal prisms positioned according to a right-handed Cartesian coordinate system and considering that the

x-,

y- and

z-axes are positively oriented to the north, the east and downward, respectively, we assume that each prism represents a different homogeneously magnetized region, with the edges coinciding with the bounds of the hook. We can estimate the magnetic moment m

^{k},

k = 1, …,

L, by comparing the synthetic data produced by the model and the vertical component of the magnetic field map measured by magnetic microscopy since the sensor-to-sample distance, the thickness of the thin section and the magnetization direction are known. Mathematically, the vertical component of magnetic field B

_{z} produced by a set of polygonal prisms at point (

x^{i}; y^{i}; z^{i}) is given by:

where

${b}_{z}^{\mathrm{ik}}$ represents the effect of the

kth prism at the

ith point (

x^{i}; y^{i}; z^{i}); x

^{k} is a vector containing the

x-coordinates of the vertices of the

kth prism; y

^{k} is a vector containing the

y-coordinates of the vertices of the

kth prism;

${\widehat{m}}^{k}$ is a unit vector in the direction of magnetization;

${m}^{k}$ is the magnetization intensity and Δ

z is the thickness of each prism. Mathematically, the vertical component of the magnetic field produced by the

kth prism is given by the following expression:

where

C_{m} is a constant that is proportional to free space permeability and

where the scalar function

${\phi}^{ik}$ is given by

and

where

${\alpha}^{k}$,

${\beta}^{k}$ and

${\gamma}^{k}$ are the Cartesian coordinates of an element inside the volume

v^{k} of the

kth 3D prism with a polygonal cross-section. This modeling is solved using a Python library Fatiando a Terra [

30]. As shown in

Figure 7, the sample was magnetized vertically with a magnetic field intensity varying from 415 mT to −31 mT. To demonstrate the applicability of the method, we use the 415 mT map (

Figure 10a). To estimate the two main contributions of the hook, we approximate the sample using three polygonal prisms with a thickness Δ

z = 30 µm and the vertices shown in

Figure 10b. We generate the vertical component of the magnetic field of the model (

Figure 10c). As we note in

Figure 10, the observed data and the synthetic data produced by modeling are very similar. The magnetic moment estimated for the head part (green prism in

Figure 10b) is

m_{head} = 1.89 × 10

^{−7} Am

^{2} and the magnetic moment for the handle part (blue prism in

Figure 10b) is

m_{handle} = 2.52 × 10

^{−6} Am

^{2}.