Enhancing Precision in Magnetic Map Interpolation for Regions with Sparse Data

Abstract

The high-precision magnetic anomaly reference map is a prerequisite for magnetic navigation and magnetic target detection. However, it is difficult to reflect the detailed characteristics of magnetic anomaly changes by using conventional data interpolation and reconstruction in the areas where magnetic anomaly gradients vary drastically and the distribution of magnetic survey lines is sparse. To solve the problem, an improved variogram of the Kriging interpolation method is proposed to improve the spatial resolution of magnetic anomaly feature. This method selects the spherical variogram model and uses the third power of the lag distance to fit the trend of magnetic anomalies. Meanwhile, the second power of the lag distance is introduced to solve the problem of under-fitting between the lag distance and the value of the variation function near the origin of the sparse variogram graph of measured data. Hyperparameter $\lambda$ is introduced to compensate for the unbalance caused by the introduction of quadratic lag in the spherical variogram model. The results of several sets of simulated and measured data show that the interpolation accuracy of the proposed method is improved by 30–50% compared with the traditional Gaussian, spherical, and exponential models in the region where the magnetic anomaly gradient changes drastically, and the proposed model provides an effective way to build a high-precision magnetic anomaly reference map of the complex magnetic background under the condition of sparse survey lines.

1. Introduction

The magnetic field is an intrinsic physical property of the Earth, which has important applications in the context of autonomous navigation and magnetic target detection, just to mention a few. With the increasingly urgent need for the high-precision autonomous navigation of underwater long-endurance unmanned submersible vehicles and submarines, the technology of magnetic navigation has become a hotspot at home and abroad [1,2,3,4,5], among which high-precision magnetic maps are the foundational and key technology of magnetic navigation [6,7]. In addition, if prior magnetic maps can be established, the accuracy of magnetic anomaly detection will be significantly improved in complex magnetic areas [8]. Therefore, as a basic work, the magnetic map is of great significance in improve navigation accuracy and target detection ability.

Aeromagnetic measurement is a means to obtain magnetic data. According to relevant measurement specifications [9,10,11,12], the magnetic data are dense in the direction of survey lines and sparse in the direction interval of survey lines. These data cannot be used for magnetic navigation and magnetic target detection directly, so it is necessary to interpolate data to establish a high-resolution and precision magnetic map. So, the algorithm of Kriging can be commonly used, which is based on the covariance function to model and predict the magnetic field space and has the advantages of higher interpolation accuracy, stronger spatial autocorrelation, and better smoothness. Wang et al. [13] compared 12 interpolation algorithms and found that the method of Kriging and radial basis function were better in generating accurate grid data. Chang et al. [14] compared the methods of Shepard surface fitting, weighted moving trend surface fitting, and others to be interpolated for gridding and verified that the accuracy of the Kriging interpolation method was better than other methods in the process of magnetic mapping. Anton et al. [15] used the Kriging algorithm to interpolate magnetic data obtained by satellites to create an instantaneous interpolation map of the magnetic field inside a tetrahedron and point out that the Kriging interpolation algorithm is worth paying close attention to for multi-point space measurements. Therefore, the Kriging algorithm can better represent the structural components of the magnetic field and accurately reflect the magnetic information in local areas, which has great advantages in complex magnetic regions [16].

By collecting the published literature on interpolation methods of magnetic data using Kriging, and considering the external usage conditions, internal fusion methods, the selection of core variogram, and other factors, it is found that relevant algorithms can be divided into the following three categories:

(1) Interpolation processing based on different Kriging algorithms.

This kind of algorithm depends on the external conditions of the Kriging interpolation algorithm. Currently, the common models include the ordinary Kriging (OK) algorithm, universal Kriging (UK) algorithm, co-Kriging (CK) algorithm, and so on. Hao et al. [17] studied the basic principle of ordinary Kriging interpolation algorithm, appropriately selected the semi-variance function model, and tested via experiment that the magnetic grid data obtained by the ordinary Kriging method were more accurate. Yang et al. [18] used cross-validation to verify that the universal Kriging algorithm has a better interpolation effect than the ordinary Kriging method. Cai et al. [19] applied the co-Kriging interpolation algorithm to indoor navigation and positioning based on magnetic field. After dividing the sample into two parts, the ordinary Kriging algorithm was carried out and then fusion processing was carried out, which can result in a small root-mean-square error and good interpolation effect.

(2) Interpolation processing based on the fusion of multiple algorithms.

In this kind of processing, other model algorithms are introduced, whose advantages are used to make up for the shortcomings of Kriging algorithm. There are many articles published on this subject. For example, Zhao et al. [20] proposed a step-by-step interpolation correction method based on the multifractal Kriging to improve the low-pass filtering characteristics of the Kriging method and reduce the loss of fractal characteristics when interpolating with sparse magnetic anomaly field data via multifractal Kriging. On the basis of this research, the step-by-step interpolation and correction method (SSICM) was proposed in [21], which conducts a singularity correction according to scale invariance in the small-scale range of the magnetic anomaly field, thereby stepwise building a reference map in gridding format. Liu et al. [22] proposed the support vector machine residual kriging method (SVMR Kriging), which first fits the magnetic trend changes using the support vector machine then interpolates the residual component via the ordinary Kriging and finally adds the two parts together to construct a regional magnetic map to improve the regional accuracy of magnetic trend changes. Zhao et al. [23] proposed a magnetic reference map construction method based on the combination of the Kriging method and the back propagation neural network (BP neural network). In the case that the interpolation error, Kriging in the region with a large gradient is greater than that of the BP neural network, and the interpolation error in the region with a small gradient is smaller than that of the BP neural network. The predicted value of the BP neural network is used to replace the interpolation result of the Kriging method in the large gradient region to improve the interpolation accuracy.

(3) Interpolation processing based on different variograms.

Variogram is the core of Kriging interpolation algorithm, which can describe not only the spatial structural changing of the regionalized variable but also its randomness changing. Commonly used variograms include the spherical variogram, exponential variogram, Gaussian variogram, and so on. Most of these papers are in the field of mathematics and geostatistics, and few are in the field of magnetic map construction. The available literature uses relevant algorithms to optimize the parameters of the variogram. For example, Li et al. [24] used the vertical integration method with the particle swarm optimization algorithm (PSO) and genetic algorithm (GA) to optimize the parameters of the variogram until the specified accuracy was reached or the maximum number of iterations was reached, thus obtaining better optimization performance and globality.

Although the interpolation algorithms based on different variograms have a certain universality, the selection or construction of what kind of variogram is more suitable for the construction of sparse magnetic maps has not been involved in relevant literatures. Therefore, based on previous studies, this paper proposes an interpolation processing method for sparse data with an improved variogram, and it uses the variogram to establish an optimal adaptive parametric interpolation method for different features of sparse data so as to improve the interpolation accuracy of sparse data and the spatial resolution of network data. At the same time, the algorithm is used to construct high-precision magnetic maps of simulation data and measured data, respectively. The results show that the interpolation accuracy of this method is better than that of Gaussian, spherical, and exponential variogram models, and the maximum absolute error, mean absolute error, and root-mean-square error are greatly reduced. It can be used to construct the local magnetic reference map with high precision and high resolution.

2. Improved Variogram of Kriging Algorithm for Sparse Data

The Kriging algorithm is a spatial interpolation method based on geostatistics, which is a minimum variance estimation and linear unbiased method for unknown data of similar characteristics in a region according to certain characteristic data of several information samples.

2.1. Principles of Kriging Interpolation Algorithm

2.1.1. Principle of Interpolation

The basic idea of the Kriging algorithm is that a certain weight is assigned to each sample value according to the spatial correlation between sample points, especially the spatial position, and the unknowns in the estimation region were estimated using the weighted average method [25].

As shown in Figure 1, $Z_i(i=1,2,\dots ,n)$ is a set of measured magnetic field sample data with coordinate position $(x_i,y_i)$, and the estimated value ${\widehat{Z}}_0$ of prediction point with coordinate position $(x_0,y_0)$ can be expressed via a linear combination of sampled magnetic field samples:

$${\widehat{Z}}_0={\displaystyle \sum }_{i=1}^n{\lambda }_i{Z}_i,$$

(1)

where ${\lambda }_i$ is the weight of the measured point $Z_i$, and ${\widehat{Z}}_0$ is called the Kriging estimation of $Z_0$. As long as the weight coefficient ${\lambda }_i$ is determined, the forecast point estimation ${\widehat{Z}}_0$ can be obtained. Kriging interpolation estimation calculates the weight ${\lambda }_i$ under the condition that the estimated ${\widehat{Z}}_0$ is unbiased and the estimation variance is minimum.

2.1.2. Variogram

The variogram is the core of the Kriging interpolation algorithm and is defined as half of the variance of the difference of the regionalized variable $Z(x)$ at two points $x,x+l$ points, i.e.,

$$\begin{array}{ll}\gamma (x,l)& =\frac{1}{2}Var\left[Z(x)-Z(x+l)\right],\\ & =\frac{1}{2}\{E{[Z(x)-Z(x+l)]}^2-{(E[Z(x)-Z(x+l)])}^2\},\\ & =\frac{1}{2}E{[Z(x)-Z(x+l)]}^2,\end{array}$$

(2)

where $\gamma (x,l)$ is the variogram and $l$ is called the lag distance.

There are more variogram models, among which the exponential variogram, Gaussian variogram, and spherical variogram are more commonly used, and their general forms are in the following order:

$$\gamma (l)=C_0+C(1-e^{-\raisebox{1ex}{$l$}\!\left/ \!\raisebox{-1ex}{$a$}\right.}),$$

(3)

$$\gamma (l)=C_0+C(1-e^{-{(\raisebox{1ex}{$l$}\!\left/ \!\raisebox{-1ex}{$a$}\right.)}^2}),$$

(4)

$$\gamma (l)=\{\begin{array}{cc}0,& l=0\\ C_0+C(\frac{3l}{2a}-\frac{l^3}{2a^3}),& 0<l\le a,\\ C_0+C& l>a\end{array}$$

(5)

where $C_0$ is the nugget, $C_0+C$ is the sill, $C$ is the arch height, and $a$ is the range.

2.1.3. Experimental Variogram

The experimental variogram ${\gamma }^{\ast }(l)$ is an estimate of the variogram $\gamma (l)$ constructed from the measured data. With the second-order stationary hypothesis, the increment of $Z(x)$ depends only on the lag $l$ between the two points, and not on the position $x$. The formula is as follows [26]:

$${\gamma }^{\ast }(l)=\frac{1}{2N(l)}{\displaystyle \sum }_{i=1}^{N(l)}{[Z(x_i)-Z(x_i+l)]}^2,$$

(6)

where $N(l)$ represents the logarithm of the data pair divided by the vector $l$. For different lags $l$, the values of ${\gamma }^{\ast }(l)$ can be calculated according to Equation (6), and points $[l_i,\gamma (l_i)]$ can be marked in the graph of the experimental variogram, which can directly reflect the relevant situation of measurement data [27].

In order to predict the unknown position of the regionalized variable, it is necessary to analyze the experimental variogram and find the best-fitting variogram model. Selecting the variogram model and parameters under optimal conditions is the key to improving the accuracy of Kriging interpolation [28].

2.2. Derivation of Variogram Formula Based on Magnetic Dipole

The magnetic map is the main tool with which to describe the distribution of the Earth’s magnetic field so as to reflect the distribution of the characteristics of the magnetic field in the target area. In general, coordinate points with the same numerical magnetic elements are connected with smooth curves to form a magnetic map, which is essentially a regular spatial arrangement of magnetic data points.

The magnetic field can be simulated by the spherical harmonic model, but for a large proportion of the magnetic field with high precision, the magnetic dipole model is usually used to simulate the magnetic field distribution of the target equivalent when the detection distance reaches 2.5~3 times the size of the target [29]. Especially in engineering practice, the target is more equivalent to the magnetic dipole model, which can also effectively approximate the far-field distribution of the target magnetic field [30]. When the magnetic dipole is located at the origin, the magnetic field of the magnetic dipole at any point $r$ in space is as follows:

$${\mathit{B}}_r=\frac{{\mu }_0}{4\pi }\left[\frac{3\langle M,r\rangle }{{\left|r\right|}^5}-\frac{M}{{\left|r\right|}^3}\right],$$

(7)

where ${\mu }_0=4\pi \times {10}^{-7}$ represents the vacuum permeability and $M$ represents the equivalent magnetic moment vector.

According to Equation (2), the variogram of two points in the simulated magnetic field based on magnetic dipoles is expressed as follows:

$$\begin{array}{ll}{\gamma }_{ij}& =\frac{1}{2}E[{\left(z_i-z_j\right)}^2]=\frac{1}{2}E[{\left|B_i-B_j\right|}^2]\\ & =\frac{{\mu }_0{}^2}{32{\pi }^2}E\left\{\frac{{\left|\left[\begin{array}{c}3{\left|r_j\right|}^5(M_x{\mathrm{r}}_x^i+M_y{\mathrm{r}}_y^i+M_z{\mathrm{r}}_z^i){\mathrm{r}}_x^i-3{\left|r_i\right|}^5(M_x{\mathrm{r}}_x^j+M_y{\mathrm{r}}_y^j+M_z{\mathrm{r}}_z^j){\mathrm{r}}_x^j+{\left|r_i\right|}^5{\left|r_j\right|}^2{M}_x-{\left|r_i\right|}^2{\left|r_j\right|}^5{M}_x\\ 3{\left|r_j\right|}^5(M_x{\mathrm{r}}_x^i+M_y{\mathrm{r}}_y^i+M_z{\mathrm{r}}_z^i){\mathrm{r}}_y^i-3{\left|r_i\right|}^5(M_x{\mathrm{r}}_x^j+M_y{\mathrm{r}}_y^j+M_z{\mathrm{r}}_z^j){\mathrm{r}}_y^j+{\left|r_i\right|}^5{\left|r_j\right|}^2{M}_y-{\left|r_i\right|}^2{\left|r_j\right|}^5{M}_y\\ 3{\left|r_j\right|}^5(M_x{\mathrm{r}}_x^i+M_y{\mathrm{r}}_y^i+M_z{\mathrm{r}}_z^i){\mathrm{r}}_z^i-3{\left|r_i\right|}^5(M_x{\mathrm{r}}_x^j+M_y{\mathrm{r}}_y^j+M_z{\mathrm{r}}_z^j){\mathrm{r}}_z^j+{\left|r_i\right|}^5{\left|r_j\right|}^2{M}_z-{\left|r_i\right|}^2{\left|r_j\right|}^5{M}_z\end{array}\right]\right|}^2}{{\left|r_i\right|}^{10}{\left|r_j\right|}^{10}}\right\}.\end{array}$$

(8)

It can be seen from the above formula that the magnetic map constructed by magnetic dipoles is not only affected by the magnetic moment in space but also has a complex relationship with the positions of two points. In order to reduce complexity, point $i$ is located directly above the magnetic dipole, point $j$ is located in the y-axis direction of the plane where the two points are located, and the plane distance between the two points is $l$. Their relative positions are shown in Figure 2. The coordinate of point $i$ is $P^i=(x_i,y_i,z_i)=(0,0,h)$, the coordinate of point $j$ is $P^j=(x_j,y_j,z_j)$$=(0,-l,h)$, and the magnetic dipole coordinate is the origin and the magnetic moment is $(0,0,M)$; then, Equation (8) can be converted to the following:

$$\begin{array}{l}{\gamma }_{ij}=\frac{{\mu }_0{}^2{M}^2}{32{\pi }^2}E\left\{\frac{(3h^4\times l{)}^2+{\left(3{\left(h^2+l^2\right)}^{5/2}-3h^5+\left(h^2+l^2\right)(h^3-{\left(h^2+l^2\right)}^{3/2})\right)}^2}{h^6{\left(h^2+l^2\right)}^5}\right\}\\ =\frac{{\mu }_0{}^2{M}^2}{32{\pi }^2}E\left\{\frac{(3{(l/\tan \phi )}^{4}l{)}^2+{\left|3{\left({(l/\tan \phi )}^2+l^2\right)}^{5/2}-3{(l/\tan \phi )}^5+\left({(l/\tan \phi )}^2+l^2\right)({(l/\tan \phi )}^3-{\left({(l/\tan \phi )}^2+l^2\right)}^{3/2})\right|}^2}{{(l/\tan \phi )}^6{\left({(l/\tan \phi )}^2+l^2\right)}^5}\right\}\\ =\frac{{\mu }_0{}^2{M}^2}{32{\pi }^2}E\left\{\frac{9{(1/\tan \phi )}^8+{\left|3{(1-\cos \phi )}^5/\sin {\phi }^5+\left({(1/\tan \phi )}^2+1^2\right)({(1/\tan \phi )}^3-{\left({(1/\tan \phi )}^2+1^2\right)}^{3/2})\right|}^2}{l^3(\cos {\phi }^6/\sin {\phi }^{16})}\right\}\\ =\frac{{\mu }_0{}^2{M}^2}{32{\pi }^2}E\left\{\frac{T}{l^3}\right\}\end{array}$$

(9)

where $T=\frac{9{(1/\tan \phi )}^8+{\left|3{(1-\cos \phi )}^5/\sin {\phi }^5+\left({(1/\tan \phi )}^2+1\right)({(1/\tan \phi )}^3-{\left({(1/\tan \phi )}^2+1\right)}^{3/2})\right|}^2}{(\cos {\phi }^6/\sin {\phi }^{16})}$, and $\phi$ is the angle of the vector ${\overrightarrow{r}}_i$ and ${\overrightarrow{r}}_j$.

As can be seen from the above equation, although the relative relationship between two points in the space of the magnetic map is relatively complex, its variogram is related to the third power of the relative position, that is, $l^3$ of the above equation. The spherical variogram is related to the third power of the lag distance $l$, and it can be seen from the constructed variogram graph that the spherical model is more in line with the trend of the variogram than other models. Therefore, the spherical variogram is chosen for research in this paper.

2.3. Construction of Improved Variogram for Sparse Data

The magnetic field information is continuous, unlike the discontinuity of geological information; although the resolution is different, the variogram graph should be continuous, especially near the origin. At the same time, the magnitude of $C_0$ is ${10}^{-10}$ by test, so the nugget $C_0$ in the traditional spherical variogram is adjusted to 0.

Secondly, it can be seen from the graph of the spherical variogram that the fitting curve of the spherical variogram is too steep and not gentle in the early stage, which is caused by the fact that the sample data carry less information at small scale, and the information bearing at large scale increases with the increase in range. For magnetic information with sparse data among the lines and dense data on the lines, the spherical variogram near the origin cannot fit the trend of lag distance and variation well because of the less-dense data samples. Therefore, this paper introduces the second power of lag distance $l$ to make it smooth near the origin, solving the problem that the lag distance near the origin of the variogram graph is over-fitted with the range due to the resolution difference caused by data sparsity.

At the same time, the introduction of the second power of the lag $l$ influences the original spherical variogram. The geometric interpretation of the spherical variogram is derived from the formula for calculating the volume of the overlapping part of two spheres with radius $a$ and spherical center distance $2h$ [31]. As mentioned above, the introduction of the second power of the lag distance $l$ must destroy the equilibrium of the volume formula. So, this paper introduces a hyperparameter $\lambda$ to control the balance between the second and third powers of the lag distance $l$. At the same time, the introduction of $\lambda$ value also provides an auxiliary parameter for regional geometric anisotropy correction.

In summary, the improved variogram model constructed in this paper is shown as follows:

$$\gamma (l)=\{\begin{array}{cc}C\left\{(1+\lambda ){\left(\frac{l}{a}\right)}^2-\lambda {\left(\frac{l}{a}\right)}^3\right\},& 0\le l\le a\\ C,& l>a\end{array}.$$

(10)

2.4. Parameters Solving of Variogram Using Nonlinear Least Squares

It can be seen from Equation (10) that the process of solving the improved variogram is transformed into solving the parameters $\lambda$, $a$, and $C$ using magnetic measurement data. The fitting problem of the improved spherical variogram model is transformed into a multiple linear regression problem, which can be solved by the Gauss–Newton nonlinear least squares algorithm. Its calculation formula is as follows:

$$\begin{array}{ll}x& =argmi{n}_{\mathsf{\Delta }x}\frac{1}{2}\parallel f(x+\mathsf{\Delta }x){\parallel }^2\\ & =argmi{n}_{\mathsf{\Delta }x}\frac{1}{2}\{f{(x)}^{T}f(x)+2f{(x)}^{T}J(x)\mathsf{\Delta }x+\mathsf{\Delta }{x}^{T}J{(x)}^{T}J(x)\mathsf{\Delta }x\}\end{array}$$

(11)

where $x={[\lambda ,a,C]}^T$,

$$\begin{array}{ll}f(x)=\{\begin{array}{ll}\gamma -C\left\{(1+\lambda ){\left(\frac{l}{a}\right)}^2-\lambda {\left(\frac{l}{a}\right)}^3\right\}\hfill & 0\le l\le a\hfill \\ \gamma -C\hfill & l>a\hfill \end{array},& J{(x)}^T=\left[\begin{array}{ccc}J{J}_{11}& J{J}_{12}& J{J}_{13}\\ \dots & \dots & \dots \\ J{J}_{n1}& J{J}_{n2}& J{J}_{n3}\end{array}\right],\\ J{J}_{11}=\{\begin{array}{ll}-C\left\{{\left(\frac{l_1}{a}\right)}^2-{\left(\frac{l_1}{a}\right)}^3\right\}\hfill & 0\le l_1\le a\hfill \\ 0\hfill & l_1>a\hfill \end{array},& J{J}_{12}=\{\begin{array}{cc}C\left\{2(1+\lambda )\frac{l_1{}^2}{a^3}-\lambda \frac{l_1{}^3}{a^4}\right\}& 0\le l_1\le a\\ 0& l_1>a\end{array},\\ J{J}_{13}=\{\begin{array}{cc}-\left\{(1+\lambda ){\left(\frac{l_1}{a}\right)}^2-\lambda {\left(\frac{l_1}{a}\right)}^3\right\}& 0\le l_1\le a\\ -1& l_1>a\end{array},& J{J}_{n1}=\{\begin{array}{ll}-c\left\{{\left(\frac{l_n}{a}\right)}^2-{\left(\frac{l_n}{a}\right)}^3\right\}\hfill & 0\le l_n\le a\hfill \\ 0\hfill & l_n>a\hfill \end{array},\\ J{J}_{n2}=\{\begin{array}{cc}C\left\{2(1+\lambda )\frac{l_n}{a^3}-\lambda \frac{l_n}{a^4}\right\}& 0\le l_n\le a\\ 0& l_n>a\end{array},& J{J}_{n3}=\{\begin{array}{ll}-\left\{(1+\lambda ){\left(\frac{l_n}{a}\right)}^2-\lambda {\left(\frac{l_n}{a}\right)}^3\right\}\hfill & 0\le l_n\le a\hfill \\ -1\hfill & l_n>a\hfill \end{array}.\end{array}$$

(12)

Take the derivative of Equation (11), and the derivative is 0.

$$\begin{array}{c}J{(x)}^{T}J(x)\Delta x=-J{(x)}^{T}f(x)\\ \Delta x=-{(J{(x)}^{T}J(x))}^{-1}J{(x)}^{T}f(x)\end{array}$$

(13)

The estimates of $\lambda$, $a$, and $C$ can be computed by iterating Equation (13), and the well-fitted improved variogram model can be obtained too.

3. Simulation

According to Chen et al. [32], the simulation of the magnetic field is obtained by using the mathematical expression of the field source with regular geometry distributed in space, such as the sphere model. This paper uses the sphere magnetic source model (referred to as the sphere combination model) to make its calculation, and the magnetic anomaly expression generated by a single-sphere magnetic source is as follows:

$$\begin{array}{ll}\Delta T\left(x,y,z\right)=& \frac{{\mu }_0}{4\pi }\frac{m}{{\left[x_1^2+y_1^2+z_1^2\right]}^{3/2}}[(2z_1^2-x_1^2-y_1^2){\sin }^{2}I+(2x_1^2-y_1^2-z_1^2){\cos }^{2}I{\cos }^{2}D+\left(2y_1^2-x_1^2-z_1^2\right){\cos }^{2}I{\sin }^{2}D\\ & -3x_1{z}_1\sin 2I\cos D+3x_1{y}_1{\cos }^{2}I\sin 2D-3y_1{z}_1\sin 2I\sin D]\end{array}$$

(14)

where ${\mu }_0$ represents the vacuum permeability, $D$ represents the declination angle of the magnetization intensity, and $I$ indicates the inclination of the magnetic field and the inclination of magnetization intensity. $m$ represents the magnetic moment, which has a relationship with the magnetic susceptibility $J$: $m=J\times \pi R^3$, and $R$ represents the radius of the sphere. $\Delta T\left(x,y,z\right)$ represents the magnetic anomaly at the position point $\left(x,y,z\right)$. The variable $\left(x_1,y_1,z_1\right)$ is calculated via the formula $x_1=x-x_0$, $y_1=y-y_0$, $z_1=z-z_0$, and $\left(x_0,y_0,z_0\right)$ represents the center coordinates of sphere.

3.1. Simulation of Magnetic Field Using Two Magnetic Dipoles

The first test: two spherical magnetic models were used to simulate the local magnetic anomaly region. The parameters are shown in

Table 1. Parameters of two spherical magnetic models.

No.	Position			Radius of the Sphere/m
	X/m	Y/m	Z/m
1	100	3000	−4000	500
2	7500	7500	4000	500

. The normal field of the magnetic background was set to 18,000 nT, and the simulated magnetic map with 50 × 50 m regional resolution was generated as shown in Figure 3a.

In order to obtain the magnetic samples on the route during the simulation of the aeromagnetic survey, the magnetic data of the simulated flight along the X direction were taken as the measured value on the constructed magnetic map, and the noise of 0.1 nT was added. The heading resolution was set to 250 m, and the flight line spacing was set to 1000 m. The schematic diagram of the measurement points is shown in Figure 3b.

The relevant parameters obtained by the method proposed in this paper are shown in

Table 2. Parameters of the improved variogram model.

Parameters of Model	$\mathit{C}$	$\mathit{a}$	$\mathit{\lambda }$
Improved model	4883.74	3305.24	0.5425

. At the same time, the magnetic map was constructed with different variogram models, as shown in Figure 4. From the results, the improved variogram method proposed in this paper has a better effect than others and greatly retains the magnetic field distribution characteristics. From the error distribution, the error of the improved variogram is distributed in the edge region due to the little information near there.

At the same time, in order to verify the effectiveness of the algorithm, we use the maximum absolute error (ME), the mean absolute error (MAE), and the root-mean-square error (RMSE) to evaluate the accuracy. ME responses are the magnitude of the error, and MAE and RMSE responses are the distribution of the error. Their formulae are as follows:

$$\begin{array}{l}\mathrm{ME}=\underset{i\in (1,\mathrm{m})}{\max }\left|z_i-{\widehat{z}}_i\right|\\ \mathrm{MAE}=\frac{1}{m}{\displaystyle \sum _{\mathrm{i}=1}^m\left|z_i-{\widehat{z}}_i\right|}\\ \mathrm{RMSE}=\sqrt{\frac{1}{m}{\displaystyle \sum _{\mathrm{i}=1}^m{\left(z_i-{\widehat{z}}_i\right)}^2}}\end{array}$$

(15)

where the parameter $m$ is the number of points in the Kriging interpolation, the parameter $z_i$ is the true value of the magnetic field, and the parameter ${\widehat{z}}_i$ is the magnetic value after interpolation.

From the comparison (as shown in

Table 3. Comparison of Kriging interpolation errors using different variogram models for the simulation of two magnetic dipoles.

Variogram Models	ME (nT)	MAE (nT)	RMSE (nT)
Gaussian Variogram	40.4315	2.9309	4.6730
Exponential Variogram	54.5584	3.8303	7.1077
Spherical Variogram	9.1834	1.7590	2.7815
Improved Variogram	4.5483	0.6291	0.7985

), it can be seen that the ME, MAE, and RMSE of the improved variogram have decreased, indicating that the algorithm proposed in this paper is better than the others.

3.2. Simulation of Magnetic Field Using Multiple Magnetic Dipoles

The second test: six spherical magnetic models were used to simulate the local magnetic anomaly region. The parameters are shown in

Table 4. Parameters of six spherical combined models.

No.	Position			Radius of the Sphere/m
	X/m	Y/m	Z/m
1	1000	7000	−1000	80
2	5000	4500	−200	63
3	4500	6000	−200	63
4	4000	4500	−250	63
5	4000	5500	−200	63
6	5000	5000	−200	14

. The Z-axis was specified as the positive direction, the normal field of the magnetic background was set to 48,000 nT, and the simulated magnetic map with 50 × 50 m regional resolution was generated as shown in Figure 5a.

In order to obtain the magnetic samples on the route during the simulation of the aeromagnetic survey, the magnetic data of the simulated flight along the Y direction were taken as the measured value on the constructed magnetic map, the heading resolution was set to 50 m, and the flight line spacing was set to 500 m. The schematic diagram of the measurement points is shown in Figure 5b.

The improved variogram model was used to fit the simulated data and then compared with the Gaussian, exponential, and spherical variogram. The comparison of the variogram graph is shown in Figure 6, and the calculation results are shown in

Table 5. Parameters and effects of the improved variogram model.

Parameters of Model	$\mathit{C}$	$\mathit{a}$	$\mathit{\lambda }$
Improved variogram	113.1403	10,157	2.01

. It can be seen from the fitting effect diagram that the improved variogram has the best continuity at the origin, which not only ensures the fitting accuracy when the lag distance is small but also indicates that the fitting effect is better than those of other models. This lays a foundation for accuracy improvement in the following process.

The magnetic map was constructed by different variogram models, as shown in Figure 7. According to the interpolation results, the improved variogram method has a better effect than the others and retains the magnetic field distribution characteristics greatly. From the error distribution, the accuracy of the improved variogram is raised by about 20 nT over the other models, and the gross errors are mainly concentrated near the magnetic anomalies.

From the comparison (as shown in

Table 6. Comparison of Kriging interpolation errors using different variogram models.

Variogram Models	ME (nT)	MAE (nT)	RMSE (nT)
Gaussian Variogram	111.1194	13.8591	29.9843
Exponential Variogram	137.775	6.7182	26.3359
Spherical Variogram	138.4306	6.8211	21.8168
Improved Variogram	68.9824	1.3386	9.6925

), it can be seen that the ME, MAE, and RMSE of the improved variogram have decreased, indicating that the algorithm proposed in this paper is optimal compared with the other three algorithms.

4. Experiments

To verify the effectiveness of this method, the aeromagnetic survey data of the Canadian Geological Survey (data sources: Kluane Lake West Aeromagnetic Survey, Residual Total magnetic Field, NTS 115G/12 and parts of NTS 115G/11, 13, 14 and NTS 115F/9 and 16, Yukon) are used for analysis and research. There are abundant deposits and significant magnetic anomalies in the Kluane area. The regional resolution of the magnetic map is 50 × 50 m, as shown in Figure 8.

The aeromagnetic sampling data were obtained along the Y direction with the distance between the measurement lines (500 m), and the noise of 0.1 nT was added. We fit these measurement data using the spherical model, exponential model, Gaussian model, and improved variogram model, respectively. The interpolation results and error distribution are shown in Figure 9. Meanwhile, the ME, MAE, and RMSE were used to evaluate the accuracy of the interpolation results (as shown in

Table 7. Comparison of Kriging interpolation errors using different variograms for measured data.

Variogram Models	ME (nT)	MAE (nT)	RMSE (nT)
Gaussian Variogram	273.94	53.55	83.96
Exponential Variogram	132.02	9.41	16.87
Spherical Variogram	131.79	10.06	17.46
Improved Variogram	61.28	5.83	10.21

). It can be seen that both the spherical and the improved variogram models can reflect the approximate position of the magnetic anomaly. However, each evaluation of the improved variogram is lower than the other models, which further proves the feasibility of the algorithm.

5. Conclusions

This work proposed an improved variogram Kriging interpolation algorithm for sparse data according to the following requirements:

• In view of the sparse characteristics of aeromagnetic data, the cubic relationship between the covariance function and the lag distance of aeromagnetic data is derived by using the magnetic dipole model, which is used as the basis for selecting the spherical variogram for interpolation processing so as to conform to the magnetic trend;
• In view of the lack of smoothness near the original point of the spherical variogram graph caused by sparse data, the quadratic lag distance is introduced for processing to solve the underfitting problem near the origin;
• In view of the imbalance of the variogram model caused by the introduction of the quadratic lag, the hyperparameter λ is introduced to deal with it.

Finally, the advantages of the improved variogram interpolation algorithm are verified by testing multiple sets of simulation data and measured data. Compared with the traditional Gaussian, spherical, and exponential models, it can improve the accuracy of data interpolation and better reflect the characteristics of the dense magnetic field in regions with drastic changes in magnetic anomaly gradient. It can greatly improve the efficiency and accuracy of constructing high-precision magnetic maps.