### Magnetic target bearing vector

A magnetic target can be considered a magnetic diploe when the distance between the target and the measurement array is greater than 2.5 times the size of the target. The magnetic field generated by the magnetic diploe cat be expressed as19

$${mathbf{B}} = frac{{mu_{0} }}{{4pi r^{5} }}[3({mathbf{M}} cdot {mathbf{r}}){mathbf{r}} – r^{2} {mathbf{M}}]$$

(1)

where µ0= 4π ×10-seven H/m is the permeability of free space. r= rr0 is the bearing vector. r0 is the unit vector of r, r is the size of r. M= mmm0 is the magnetic moment of the magnetic diploe. M is the size of M, m0 is the unit vector of M . “. ” presents the dot product of the vectors.

The magnitude of Bcan be calculated as

$$left| {mathbf{B}} right| = frac{mu Ms}{{4pi r^{3} }}$$

(2)

where s =[1 + 3(t)2]1/2 and you= r0m0. The magnitude ofBcan be measured by a scalar magnetometer. As can be seen in Eq. (2), |B | can be seen as a rotational invariant20. | B| understands only the magnitude ofr and magnetic target bearing information cannot be obtained. In order to understand the bearing information of the magnetic target, furthermore, the parameter s is assumed to be a spatial constant, we calculated the gradient of | B| as following

$${mathbf{G}} = nabla left| {mathbf{B}} right| = – frac{3mu Ms}{{4pi r^{4} }}{mathbf{r}}_{{mathbf{0}}}$$

(3)

and the magnitude ofgis

$$left| {mathbf{G}} right| = frac{3mu Ms}{{4pi r^{4} }}$$

(4)

According to Eqs. (3) and (4), the unit bearing vector can be calculated as

$${mathbf{r}}_{{mathbf{0}}} = – frac{{mathbf{G}}}{{left| {mathbf{G}} right|}}$$

(5)

According to Eqs. (2) and (4), the magnitude of the bearing vector can be obtained

$$r = 3frac{{left| {mathbf{B}} right|}}{{left| {mathbf{G}} right|}}$$

(6)

Therefore, the bearing vector can be expressed as

$${mathbf{r}} = r cdot {mathbf{r}}_{{mathbf{0}}} = – frac{{3left| {mathbf{B}} right|}}{{left| {mathbf{G}} right|^{2} }}{mathbf{G}}$$

(seven)

Equation (7) shows that the bearing vector can be calculated by total field | B| and its gradient datag . | B| andg both can be measured by an array of scalar magnetometers. But, in fact, the parameters is not a spatial constant. Under these conditions, the gradient of | B | is deducted as

$${mathbf{G}} = nabla left| {mathbf{B}} right| = frac{3mu Ms}{{4pi r^{4} }}[k_{2} {mathbf{r}}_{{mathbf{0}}} – k_{1} {mathbf{m}}_{{mathbf{0}}} ]$$

(8)

where k1=you / s2, k2= you2/ s2+ 1. Then the magnitude of gcan also be obtained

$$left| {mathbf{G}} right| = frac{3mu Ms}{{4pi r^{4} }}k_{3}$$

(9)

where k3=[1 + (1 − t2)t2/s4]1/2. To obtain the unit bearing vector r0we define the unit vector of gfrom eqs. (8) and (9).

$${mathbf{G}}_{{mathbf{0}}} = frac{{mathbf{G}}}{{left| {mathbf{G}} right|}} = frac{{k_{2} {mathbf{r}}_{{mathbf{0}}} – k_{1} {mathbf{m}} _{{mathbf{0}}} }}{{k_{3} }}$$

(ten)

Therefore, the unit bearing vectorr0 can be deduced from Eq. (ten)

$${mathbf{r}}_{{mathbf{0}}} = frac{{k_{1} {mathbf{m}}_{{mathbf{0}}} + {mathbf{ G}}_{{mathbf{0}}} k_{3} }}{{k_{2} }}$$

(11)

The magnitude of the bearing vector can be obtained from the equations. (2) and (9)

$$r = frac{{left| {mathbf{B}} right|}}{{left| {mathbf{G}} right|}}k_{3}$$

(12)

The bearing vector can be expressed as

$${mathbf{r}} = r. {mathbf{r}}_{{mathbf{0}}} = frac{{k_{3} left| {mathbf{B}} right|}}{{left| {mathbf{G}} right|}}frac{{k_{1} {mathbf{m}}_{{mathbf{0}}} + {mathbf{G}}_{{mathbf {0}}} k_{3} }}{{k_{2} }}$$

(13)

### Iteration algorithm

As can be seen in Eq. (13), the bearing vector is determined by k1 , k2 , k3total field | B| and its gradient data g . If we know m0 andr0, k1 , k2 and k3can be calculated by their definition, total field | B| and its gradient data gcan be measured by an array of scalar magnetometers. Generally, the magnetic field of the magnetic target is aggravated by a hard and induced magnetic field21. When the hard magnetic field is small and the induced magnetic field is the main component, the orientation of the magnetic moment of the target is parallel to the orientation of the geomagnetic field. In practice, the induced magnetic field is much more important than the hard magnetic field18. In this study, only the induced magnetic field is considered. Thus, the unit magnetic moment vector m0 can be calculated based on magnetic inclination and declination, m0 can be expressed as follows

$${mathbf{m}}_{{mathbf{0}}} = [cos (I)cos (D),cos (I)sin (D),sin (I)]$$

(14)

where, Iand Drepresent respectively the magnetic inclination and declination of the local geomagnetic field at the measurement position. Their values ​​can be obtained from the International Geomagnetic Reference Field (IGRF).

r0 and m0 can be calculated by Eqs. (5) and (14), respectively, therefore,k1,k2andk3can be calculated. However, the r0 calculated by eq. (5) is an approximation because the parametersis assumed to be a spatial constant. In order to improve the accuracy of the bearing vector, an iteration algorithm is used to update ther0 and the iteration algorithm pseudocode is expressed in Table 1.

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