Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 2, No. 1, 2022 

 

103 

Morphological	Inversion	Initial	Model	Position	
Estimation	Method	
Ziming Cai, Zhining Li 

Shijiazhuang campus, Army Engineering University of PLA, Shijiazhuang 050003, Hebei, China 

 

Abstract: The shape inversion method can clearly invert the three-dimensional attitude of the magnetic target, but the shape 
inversion needs to obtain the position information of the magnetic target to be measured. In order to better determine the magnetic 
target position information, a magnetic target morphology inversion method is proposed, which fuses the physical property 
inversion results. The center position of the magnetic target is calculated by the physical property inversion method; the initial 
model is set, the module growth selection function is established by using the L2 norm and the distance constraint function, and 
the optimal growth module in the morphological inversion process is selected; through continuous iterative update, the magnetic 
The 3D pose of the target. The position error calculated by this method is less than 10% of the actual value, and the inversion 
coincidence rate is greater than 80%. The results show that this method can effectively improve the accuracy of morphology 
inversion. 

Keywords: Shape inversion, Physical property inversion, Magnetic targets, Magnetic gradient tensor, Prior. 
 

1. Introduction 
Magnetic target inversion is a method of inverting the 

magnetic parameters of the magnetic target through the 
magnetic survey data and using the obtained results to infer 
the information such as the shape and position of the magnetic 
target. Magnetic target inversion can intuitively provide 
magnetic target feature information, and it has very important 
application value in the fields of military reconnaissance, 
UXO removal and geological exploration [1]-[7]. 

The inversion methods of magnetic targets are mainly 
divided into physical property inversion and morphology 
inversion [8]. Physical property inversion is based on the 
magnetic field relationship between each point in the space to 
be measured and each point on the measurement surface, 
using the measurement data to calculate the magnetic 
parameters of each point in the space to be measured, and 
determining the three-dimensional attitude of the magnetic 
target based on the magnetic parameter value of each point 
[9] ]-[11]. Because the number of points in the space to be 
measured is far more than the number of measurement points, 
the solution of physical property inversion is 
underdetermined, and the solution is not unique and unstable, 
resulting in unclear inversion results [12]. Morphological 
inversion is to continuously fit the corresponding geometric 
shape through the observation data under certain prior 
conditions, and simulate the three-dimensional shape of the 
magnetic target through the fitted geometry [13]-[14]. The 
morphology inversion method does not need to solve the 
inversion equation, avoids the ill-conditioned multi-solution 
problem in the inversion process, and can invert the attitude 
of the magnetic target more clearly, but its inversion accuracy 
depends on the prior information of the magnetic target to be 
measured [ 15]. 

In order to improve the accuracy of the morphological 
inversion method, this paper proposes a morphological 
inversion method of the magnetic target based on the physical 
property inversion results. The information sets the initial 
model, selects the appropriate module growth function to 
make the model grow and iterates, and finally obtains the 

three-dimensional pose of the magnetic target. 

2. Method 
The three-dimensional inversion process of the magnetic 

target in the morphological inversion method is as follows: 
first, use the magnetic gradient tensor data measured on the 
observation surface to calculate the center position of the 
magnetic target, and select the gridded area to be measured 
according to the calculated prior conditions. A cuboid mesh is 
used as the initial model in the morphological inversion, and 
it is given corresponding magnetic parameters; then a 
growing area is established around the initial model, and the 
growing area is composed of a cuboid that coincides with at 
least one face of the current model; According to certain 
judgment criteria, the optimal growth module is selected, and 
the current model is iterated to realize the construction of the 
inversion model; when the iteration termination condition is 
reached, the model stops iterating, and the three-dimensional 
pose of the magnetic target is obtained. 

2.1. Morphological Inversion Prior Parameter 
Calculation 

2.1.1. Parameter Calculation Based on Physical Property 
Inversion 

Physical property inversion is to grid the space to be 
inverted, and establish a nonlinear equation between each grid 
and measurement points, and then solve the physical 
parameters of each grid. The generated magnetic outliers are 
equal to the measured values, that is, the cumulative value of 
the magnetic outliers generated by each grid in the space to 
be inverted for a certain measurement point should be equal 
to the measured value, which can be expressed as: 

 
𝐺 ∑ 𝐴 𝑚                 (1) 

 
In the formula, Gi is the measurement value of the i-th 

(where i=1,…,N,N is the number of measurement points); mj 
is the j-th (j=1,…,M, M is the grid number) net The physical 
parameters of the grid, this paper is the magnetization; Aij is 



 

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the kernel function of the i-th measurement point 
corresponding to the j-th grid. Write equation (1) in matrix 
form: 

 
𝑮 𝑨 ∙ 𝒎 𝑀 ≫ 𝑛)                (2) 

 
In the formula, G is the measurement point data matrix, A 

is the kernel function matrix, and m is the magnetic target 
physical property parameter matrix. 

It can be seen from formula (2) that the number of grids is 
much larger than the number of measurement points, which 
results in the equation being an underdetermined equation 
with multiple solutions. Therefore, the Tikhonv regularization 
method is used, and the regular term is introduced to limit the 
range of the solution, and the equation is transformed into the 
least squares form: 

 
𝛼 min ‖𝑮 𝑨 ∙ 𝒎‖ 𝜆 ‖𝐿 ∙ 𝒎‖         (3) 

 
where α is the objective function, Lij=ωj∙δij, ωij is the depth 

weighting function, δij is the Kronecker function, and λ is the 
regularization parameter. In the expression of the depth 
weighting function ωj, α is the objective function, Lij=ωj∙δij, 
ωij is the depth weighting function, δij is the Kronecker 
function, and λ is the regularization parameter. The 
expression of the depth weighting function ωj is: The formula 
is: 

 
𝜔                   (4) 

 
In the formula, zj is the center depth of the jth grid, and ξ>0 

can prevent the depth weighting function from generating 
singularities. 

The physical property inversion can obtain the 
magnetization of each grid. According to the magnetization 
value of each grid, the boundary and shape of the magnetic 
target can be determined, and the horizontal boundary and 
depth information of the magnetic target can be provided for 
the subsequent shape inversion. 

2.2. Selection of the Optimal Growth Module 
for Morphological Inversion 

Morphological inversion is to fit the three-dimensional 
attitude of the magnetic target through the growing model. 
First, the position information and magnetic parameters of the 
magnetic target are calculated, and then the initial model is 
set according to the position information, and the 
corresponding magnetic parameters are assigned to build 
around the current model. For the region to be grown, the 
optimal growth module is selected according to the magnetic 
measurement data, and the model continues to iteratively 
grow until the termination condition is reached, and the final 
inversion result is the obtained three-dimensional pose of the 
magnetic target. 

In the process of model growth, the optimal growth module 
needs to be selected in the area to be grown. Therefore, a 
module growth selection function is established (the 
minimum value of the function is taken as the optimal value): 

 
𝛤 𝑚 ∑ 𝛾 𝑚 𝜌𝜃 𝑚           (5) 

 
In the formula, m is the magnetic parameter of the magnetic 

target; R is the number of data types; ρ is the regularization 
parameter. Since θ(m) will increase with the model growth, 
the constant decaying regularization parameter ρ is set for 
balance. In this paper, ρ Take an adaptive approach: 

 
𝜌 a𝜌                    (6) 

 

where n represents the number of iterations, setρ 1, 
a 0.95. 

In formula (6), γ(m) is the L2 norm of the measured data 
and the model forward data, which ensures that the difference 
between the model forward data and the measured data 
gradually decreases, so that the growth model is closer to the 
actual shape of the magnetic target; θ(m ) prevents the model 
from growing in a certain direction by limiting the distance 
between the new module and the original model. The specific 
formulas of the above three constraint functions are: 

 
𝛾 𝑚 ζ𝑮 𝑨𝑚              (7) 

 
θ 𝑚 ∑ L             (8) 

 
In the formula, ζ is the scale factor, and Am can be 

calculated by the forward method; x, y, and z represent the 
lengths of the current model in the x, y, and z axes, 
respectively, W is the number of modules in the current model, 
and Lp is the pth module. The distance from the center of the 
newly added module. 

2.3. Establishment of Iterative Termination 
Conditions for Morphological Inversion 

The update of the model is continuous, so it is necessary to 
set a certain iterative termination condition so that the model 
stops updating when the inversion is completed. Using the 
module growth selection function, the iteration termination 
condition can be constructed, and its expression is: 

 

Ε 𝑚
|   |

|  |
            (9) 

 
In the formula, Υ  𝑚   is the value when the optimal 

growth module is not added, and Υ  𝑚  is the value after 
adding the optimal growth module. When the iteration 
termination function value is less than a certain threshold, it 
indicates the effect of the new module on the fitting. not much, 
so terminate the iteration. 

3. Simulation Analysis 
In order to verify the influence of the prior conditions on 

the morphological inversion results and the applicability of 
the method in this paper, the following simulation analysis is 
carried out. Set up two sets of magnetic targets with different 
shapes for simulation, which are cuboid magnetic targets with 
length, width and height of 110m, 10m and 6m and slender 
magnetic targets with length, width and height of 2m, 2m and 
12m, as shown in Figure 1. Show. Set the center position of 
the slender model to (0m, 0m, 9m), the magnetization is 
40A/m, the magnetization direction is I=17°, D=50°; the 
center position of the cuboid model is (0m, 0m, 9m), The 
magnetization is 40A/m, and the magnetization direction is 
I=25°, D=65°; the model forward data is used as the simulated 
measurement data, as shown in Figures 2 and 3. 



 

105 

 
 

 

 
Figure 1. Schematic Diagram of the Slender Model (Left) and the Flat Model (Right) 

 

 
Figure 2. Forward Calculation Data of the Slender Model 

 



 

106 

 
Figure 3. Forward Calculation Data of Cuboid model 

 
Using the method in this paper to calculate the prior 

information of the two groups of models respectively, the 
calculation results show that the center position of the Cuboid 
model is (0m, 0m, 10m), and the center position of the slender 
model is (0m, 0m, 9m). The calculation results are shown in 
Figure 4. 5 shown. Use the obtained results to perform 
morphological inversion. In order to avoid being affected by 

the magnetization direction, the Bzz data that is insensitive to 
the magnetization direction is used for the inversion. Since 
the magnetization has little effect on the inversion results, it 
is not calculated separately and is set to 20A/ m, the inversion 
results are shown in Fig. 6. The inversion results of the two 
models and the RMSE statistics of the measured data are 
shown in Table 1. 
 

 
(a)                            (b)                           (c) 

Figure 4. Estimation Results of Prior Information of Slender Magnetic Target Simulation Data 
(a) y=0 Direction Slice; (b) x=0 Direction Slice; (c) z=8 Direction Slice 



 

107 

 
(a)                                   (b)                               (c) 

Figure 5. Estimation Results of Prior Information of Cuboid Simulation Data 
(a) y=0 Direction Slice; (b) x=0 Direction Slice; (c) z=9 Direction Slice 

 

 
Figure 6. Inversion Results of Slender Model(left)and Cuboid Model(right) 

 

Table 1. The RMSE Statistics of the Different Shape Models’ Inversion Results and the Measured Data 

Magnetic target shape 
Bxx 
nT/m 

Bxy 
nT/m 

Bxz 
nT/m 

Byy 
nT/m 

Byz 
nT/m 

Bzz 
nT/m 

Cuboid Model 24.8166 12.2043 28.5468 21.5280 25.7548 38.5796 
Slender Model 18.2684 10.5684 22.2684 18.5217 23.5464 32.1258 

 
It can be seen from the results in Figure 6 and Table 1 that for magnetic targets of various shapes, this method can 



 

108 

perform inversion and obtain relatively accurate results. 

4. Conclusion 
Aiming at the problems existing in the inversion process of 

the magnetic target shape, this paper proposes a shape 
inversion method that integrates the physical property 
inversion results. Then, set the parameters of the initial model, 
use the module growth selection function to select the optimal 
growth module, iteratively update the inversion model, and 
finally fit the three-dimensional pose of the magnetic target. 
The simulation and experimental results can prove the 
following conclusions: 

(1) The method in this paper can accurately calculate that 
the error of the position information of the magnetic target is 
less than 10% of the actual value of the magnetic target, which 
improves the accuracy of the morphology inversion; and with 
the improvement of the physical property inversion method, 
the accuracy of the method in this paper is further increased. . 

(2) For magnetic targets with different shapes, the method 
in this paper can determine the shape type by calculating the 
prior information, so as to adjust the module growth selection 
function and improve the applicability of the shape inversion 
method. 

Using the physical property inversion results as prior 
information to perform morphological inversion can improve 
the coincidence rate between the inversion results and real 
objects, but due to the limitation of mesh division density, in 
order to ensure the calculation efficiency, the mesh density 
cannot be set too high, resulting in computational As a result, 
the resolution is degraded and a certain error occurs. 
Therefore, further optimization will be carried out to address 
this issue in the follow-up research. 

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