## 2 Forward and Inversion Theory
### Magnetic Three-Dimensional Inversion
#### 2.2.2 Three-Dimensional Limited-Memory BFGS Inversion Using a Magnetic Method  
In this paper, a three-dimensional limited-memory BFGS (L-BFGS) inversion method based on the multiplicative objective function is proposed. This inversion method uses the log-barrier term omitted by the logarithm of model terms, while the multiplicative objective function eliminates the problem of regularization factor selection required by the traditional inversion method. At the same time, the limited-memory BFGS method has a relatively high inversion efficiency.  
The objective function of the three-dimensional finite memory quasi-Newtonian inversion by the magnetic method based on the multiplicative objective function is [17]:  
\[\Phi(m)=\Phi_{d}\times\Phi_{m_{2}}, \tag{8}\]  
\[\Phi_{d}=\big{(}d-Gm\big{)}^{T}C_{d}^{-1}\big{(}d-Gm\big{)}, \tag{9}\]  
\[\Phi_{m_{2}}=\big{(}m-m_{pri}\big{)}^{T}C_{m}^{-1}\big{(}m-m_{pri}\big{)}+\delta, \tag{10}\]  
where \(\Phi_{d}\) is the data item, \(\Phi_{m_{2}}\) is the model item, \(m\) is the permeability \(\kappa\), \(m_{pri}\) is the prior model vector, G is the kernel function matrix, \(C_{m}\) is the model covariance matrix, and \(C_{d}\) is the data covariance matrix, \(T\) is the transpose operation.  
The gradient of the objective function is:  
\[\psi=\Phi_{d}\times 2C_{m}^{-1}\big{(}m-m_{pri}\big{)}+2C_{d}^{-1}G^{T}(d-Gm) \times\Phi_{m_{2}}, \tag{11}\]  
where \(G^{T}\) is the transpose of the kernel matrix.  
In addition, \(\delta\) in Equation (10) is a constant. To prevent the first inversion of the model term being 0, the objective function is set to 0, resulting in inversion failure.  
The method for selecting the constant \(\delta\)[19] is as follows:  
\[\delta=\sqrt{\frac{\mathrm{N}}{\Delta\mathrm{V}}}\, \tag{12}\]  
where N is the total number of observed data and \(\Delta\mathrm{V}\) is the average volume of grid division units. In this paper, the uniform mesh is divided, so \(\Delta\mathrm{V}\) is the volume of any cube unit.  
L-BFGS uses the objective function gradient of the first (3-20) iterations to approximate the Hessian matrix, which is more efficient and faster than the BFGS and Gauss-Newton methods.  
The L-BFGS inversion procedure is as follows:  
1. Determine the initial inversion model, set the maximum number of inversion iterations and the minimum fitting difference, calculate the model item and add the constant value, and carry out depth weighting on the initial model;
2. Take the natural logarithm of the initial model \(m\) and \(m_{pri}\);
3. Calculate the objective function \(\Phi\) and the gradient \(\psi\);
4. Calculate the inverse matrix \([H_{k}]^{-1}\) and the quasi-  
The initial step size of the first and subsequent iterations of the inversion is selected as follows:  
\[\alpha_{1}=\frac{\text{P}}{\parallel\psi\parallel_{2}}, \tag{19}\]  
\[\alpha_{2,3...}=1, \tag{20}\]  
where \(\alpha_{1}\) is the initial step of the first iteration, \(\alpha_{2,3...}\) is the initial step of subsequent iterations, and the most appropriate value of p ranges from 5 to 15.  
In addition, the termination condition of the inversion is that the data fitting difference is less than the predefined minimum fitting difference or that the number of iterations reaches the set value. The formula for calculating the data fitting difference (\(RMS\)) is as follows:  
\[RMS=\sqrt{\frac{(d-Gm)^{T}C_{d}^{-1}(d-Gm)}{N}}, \tag{21}\]  
where N is the total number of observed data.