Journal of Sensors

Volume 2018, Article ID 5798287, 7 pages

https://doi.org/10.1155/2018/5798287

## Aeromagnetic Compensation Algorithm Based on Principal Component Analysis

^{1}Key Laboratory of Electromagnetic Radiation and Sensing Technology, Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China^{2}University of Chinese Academy of Sciences, Beijing 100049, China

Correspondence should be addressed to Peilin Wu; moc.liamxof@429gnolwuw

Received 26 May 2017; Revised 19 December 2017; Accepted 4 January 2018; Published 8 April 2018

Academic Editor: Jesus Corres

Copyright © 2018 Peilin Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Aeromagnetic exploration is an important exploration method in geophysics. The data is typically measured by optically pumped magnetometer mounted on an aircraft. But any aircraft produces significant levels of magnetic interference. Therefore, aeromagnetic compensation is important in aeromagnetic exploration. However, multicollinearity of the aeromagnetic compensation model degrades the performance of the compensation. To address this issue, a novel aeromagnetic compensation method based on principal component analysis is proposed. Using the algorithm, the correlation in the feature matrix is eliminated and the principal components are using to construct the hyperplane to compensate the platform-generated magnetic fields. The algorithm was tested using a helicopter, and the obtained improvement ratio is 9.86. The compensated quality is almost the same or slightly better than the ridge regression. The validity of the proposed method was experimentally demonstrated.

#### 1. Introduction

Aeromagnetic exploration is one of the most important exploration methods used in geophysical exploration, especially in mineral exploitations and underground unexploded ordnance detections [1–3]. Compared with surface magnetic survey, aeromagnetic exploration is a high efficiency and security method. Prospectors can finish a large survey area in a short time without any risk from the dangerous areas. Besides, the data measured by optically pumped magnetometer (OPM) is typically reliable and accurate [4].

The platform for aeromagnetic exploration is an aircraft like helicopter. Considering that the construction materials always contain some ferromagnetic materials, aircrafts can generate magnetic interference fields. The platform-generated fields degrade the measurement accuracy of the OPM, and this has necessitated the application of aeromagnetic compensation.

Aeromagnetic researches can trace back to World War II. In order to detect the submarine under water, U.S. researchers developed an aeromagnetic system. After the war, because of its excellent application in geophysics, aeromagnetic was widely used in the civilian field. Meantime, aeromagnetic compensation methods were widely researched. The specific compensation model was proposed by Tolles in the 1940s [5]. He also designed a hardware compensation system [6]. After that, various compensation methods and systems were presented [7–13]. For instance, Leliak proposed sinusoidal maneuvers for compensating the platform-generated fields [14]. Leach proposed a ridge regression algorithm to solve multicollinearity in the aeromagnetic compensation model [15]. Leach’s method can obtain good results in aeromagnetic compensation. But the method needs to calculate an optimal ridge coefficient. Sometimes it is difficult to calculate this optimal ridge coefficient.

To address multicollinearity in the model, we propose a novel method based on principal component analysis (PCA) to compensate platform-generated fields. This method uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables [16, 17]. After transformation, multicollinearity of the model is eliminated, and thus least squares (LS) algorithm becomes more accurate. Compared with the ridge regression algorithm, the proposed method has almost the same or slightly better performance. Meanwhile, the proposed method is not necessary to calculate the optimal ridge coefficient and this makes it easier to be used in practice.

#### 2. Aeromagnetic Compensation Model

##### 2.1. Mathematical Expression of Aeromagnetic Compensation Model

Aeromagnetic compensation involves the suppression of the platform-generated fields. In exploration, the magnetic fields are measured by OPM, and the attitude angels of the platform are measured by a vector magnetometer. The direction cosines of the attitude angels of the platform are calculated by the output of the vector magnetometer as follows: where , , and are, respectively, the direction cosines of the attitude angels of the platform, and , , and are, respectively, the transverse, longitudinal, and vertical components of Earth’s total magnetic field measured by the vector magnetometer.

In the aeromagnetic compensation model, the platform-generated fields can be grouped into three types: permanent, induced, and eddy-current magnetic fields. The permanent magnetic fields caused by the permanent magnetism of the ferromagnetic parts of the platform can be given by where denotes the compensation coefficients of the permanent magnetic fields and denotes the variables constituted by the direction cosines.

The induced magnetic fields created in paramagnetic parts by the Earth’s magnetic field can be given by where is the Earth’s magnetic field which can be calculated by ; denotes the compensation coefficients of the induced magnetic fields; and denotes the variables consisting of the direction cosines and .

The eddy-current magnetic fields produced by eddy currents that occur in any electrical conducting path of the airframe can be given by where , , and are, respectively, the time derivative of , , and ; denotes the compensation coefficients of the eddy-current magnetic fields; and denotes the variables consisting of the direction cosines and .

The platform-generated fields measured by OPM can be expressed as

The corresponding matrix notations of (5) are expressed as where and are column vectors consisting of and ; and is a feature matrix can be written as

The LS algorithm is used to get an optimum solution of (6). It can be expressed as

Because multicollinearity existed in the model, the inverse matrix of does not exist; therefore, (8) cannot obtain the accurate compensation coefficients.

Considering the issue, by adding regularization term to the loss function of the LS algorithm, the new loss function can be given by where is a ridge parameter. When equals zero, (9) becomes the loss function of LS algorithm. By equating the partial derivative of (9) with respect to to zero, the expression can be given by

The solution of (10) is

Compared with (8) in aeromagnetic compensation, (11) leads to better compensation results. In a typical case, using (11) in a calibration flight, the compensation coefficients can be obtained. Then by plugging the obtained compensation coefficients back into (6), the platform-generated fields can be fitted and compensated.

##### 2.2. Multicollinearity of the Aeromagnetic Compensation Model and Assessment Criteria of Multicollinearity

Aeromagnetic compensation model is a linear model. In (6), multicollinearity exists between the variables , which are the column vectors of matrix . The relationship can be expressed as where parameters are not all zeros and is a constant [18].

In (6), matrix is calculated by the different combinations of the direction cosines and its time derivative. Therefore, multicollinearity widely exists in the aeromagnetic compensation model. The typically multicollinearity relationships are derived from (3) and (4). They can be expressed as follows:

Variance inflation factor (VIF) is used to quantify the severity of multicollinearity in the model [19]. First, the sum of squares of residuals can be expressed as follows:
where is the *j*th sampling point of the variable and is the *j*th predicted value associated with .

Then, the total sum of squares can be given by where is the mean of the observed data.

After that, the coefficient of determination of the *i*th variable can be defined as

From (16), the mathematical expression of VIF can be given by

If the VIFs of the variables of the calibration data are larger than 10, it means that multicollinearity is significant in the model. The LS algorithm cannot obtain accurate compensation results. Thus, PCA is introduced to eliminate multicollinearity in the model. After that, accurate compensation results are obtainable.

#### 3. Aeromagnetic Compensation Method Based on PCA

PCA is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components (PCs). Using appropriate PCs, the overlap of the useful information in matrix caused by multicollinearity can be addressed. Meantime, this method is effective to separate the noise and the useful information and improve the accuracy of aeromagnetic compensation.

In the aeromagnetic compensation model, the PCA extracts PCs from the variables , where . After PCA, the feature matrix is converted into a new feature matrix , which consists of column vectors . If there is no multicollinearity in the model, the matrix and have same number of columns; if multicollinearity exists, the number of columns of is less.

##### 3.1. Process of the Method

The structure of the aeromagnetic compensation method based on PCA is shown in Figure 1.