Aeromagnetic Compensation Algorithm Based on Principal Component Analysis

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.


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][2][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 platformgenerated 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][8][9][10][11][12][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.

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 u 1 , u 2 , and u 3 are, respectively, the direction cosines of the attitude angels of the platform, and T t , L t , and V t 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 platformgenerated 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 c i , i = 1, … , 3, denotes the compensation coefficients of the permanent magnetic fields and A i , i = 1, … , 3, 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 H e t is the Earth's magnetic field which can be calculated by T t 2 + L t 2 + V t 2 ; c i , i = 4, … , 9, denotes the compensation coefficients of the induced magnetic fields; and A i , i = 4, … , 9, denotes the variables consisting of the direction cosines and H e t .The eddy-current magnetic fields produced by eddy currents that occur in any electrical conducting path of the airframe can be given by where u 1 ′, u 2 ′, and u 3 ′ are, respectively, the time derivative of u 1 , u 2 , and u 3 ; c i , i = 10, … , 18, denotes the compensation coefficients of the eddy-current magnetic fields; and A i , i = 10, … , 18, denotes the variables consisting of the direction cosines and H e t .
The platform-generated fields measured by OPM can be expressed as The corresponding matrix notations of ( 5) are expressed as where H d and C are column vectors consisting of H d t and c i , i = 1, … , 18; and A 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 A T A does not exist; therefore, (8) cannot obtain the accurate compensation coefficients.
Considering the issue, by adding L 2 regularization term to the loss function of the LS algorithm, the new loss function can be given by where α is a ridge parameter.When α equals zero, (9) becomes the loss function of LS algorithm.By equating the partial derivative of ( 9) with respect to C to zero, the expression can be given by The solution of ( 10) is Journal of Sensors 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.

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 A 1 , A 2 , … , A 18 , which are the column vectors of matrix A. The relationship can be expressed as where parameters b j are not all zeros and c 0 is a constant [18].In (6), matrix A 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 A i j is the jth sampling point of the variable A i and Âi j is the jth predicted value associated with A i j .Then, the total sum of squares can be given by where A i j is the mean of the observed data.
After that, the coefficient of determination of the ith variable can be defined as From ( 16), the mathematical expression of VIF can be given by If the VIFs of the variables A 1 , A 2 , … , A 18 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.

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 A 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 m PCs F 1 , F 2 , … , F m from the variables A 1 , A 2 , … , A 18 , where m ≤ 18.After PCA, the feature matrix A is converted into a new feature matrix F, which consists of column vectors F i .If there is no multicollinearity in the model, the matrix F and A have same number of columns; if multicollinearity exists, the number of columns of F is less.
3.1.Process of the Method.The structure of the aeromagnetic compensation method based on PCA is shown in Figure 1.
The specific steps are as follows: (1) The feature vectors A 1 cal , A 2 cal , … , A 18 cal obtained by calibration flight data should be standardized as where A i cal is the ith column vector of the standardized calibration flight matrix A cal ; A i cal is the ith column vector of the calibration flight matrix A cal ; A i cal and s i cal are, respectively, the mean and standard deviation of the corresponding observed data.
(2) Calculate the covariance matrix of the matrix A cal and denote it as V.
(4) The ith PC can be expressed as Then rank the PCs by their contributions.The new calibration flight matrix consisting of PCs can be denoted as where m ≤ 18.Thus, the relationship between A cal and F cal can be given by where a is the matrix consisting of the corresponding eigenvectors.

Journal of Sensors
Using the calibration flight data, the aeromagnetic compensation coefficients can be calculated by LS algorithm, which is expressed as (5) Using the mean and standard deviation of the matrix A cal , the corresponding test flight matrix A test can be standardized as where F test is the new test flight matrix.Then the compensation data can be obtained by where H OPM test is the data measured by OPM before compensation; and H Comp test is the data after compensation.
The optimal number of the the PCs is determined through assessing quality of the compensated data.Then the optimal compensation hyperplane is obtained.

Assessment Criteria for the Number of PCs.
The key point in the proposed method is to choose the PCs.Too few PCs cannot compensate the platform-generated fields, while too many PCs cannot eliminate multicollinearity in the aeromagnetic compensation model.Therefore, it is necessary to choose appropriate number of PCs.
The test flight is used to address this issue.First, the PCs are ranked by their contributions to the model.Then in this order, the PCs enter the model in sequence.Meantime, the improvement ratios (IR) are assessed by [20] where σ u is the standard deviation of the uncompensated data; and σ c is the standard deviation of the compensation data.If the IR obtains maximal value, the number of the PCs in the model is optimal.

Results and Discussion
4.1.Helicopter Experiment.A helicopter mounted with a straight rod was used to carry out an experiment to demonstrate the validity of the method.The helicopter and its experimental flight path are shown in Figure 2. The OPM is mounted at the front of the straight rod.
In order to avoid being disturbed by the magnetic interference of the ground, the experimental height was about 3000 m.First the helicopter flew a calibration box, and then Both of them contain four orthogonal magnetic headings, and each line of the boxes contains calibration maneuvers consisting of ±10 ∘ rolls, ±5 ∘ pitches, and ±5 ∘ yaws.In ideal calibration flight, the calibration box and the test box are in the same shape.However, in practical flight, it is difficult to keep these two boxes in the same shape, for the impact of airflow, operation accuracy of the pilot, and some other reasons like accuracy of flight control system.However, these unsatisfactory factors have litter effect on the performance of aeromagnetic compensation.The calibration flight is used to obtain a compensation hyperplane, and the test flight is used to assess the performance of the compensation hyperplane.
After the compensation hyperplane is assessed, the optimal number of the PCs is obtained.The original measurement data are shown in Figure 3.

Quantitative Analysis of the Experimental Results
. The multicollinearity analysis is applied to the calibration flight data before compensation.The VIFs of the variables of the calibration flight data are listed in Table 1.
In Table 1, all VIFs of variables are greater than 10, and the phenomenon means that multicollinearity is obvious in the model.Using the method based on PCA, multicollinearity of the model is eliminated.And all VIFs of the new variables are equal to one.
The relationship between the IRs and the numbers of the PCs is shown in Figure 4.The vertical axis is the value of IR and the horizontal axis is the number of the PCs entering the model.
Figure 4(b) shows that the maximal IR is obtained when the number is 14.It means that the corresponding hyperplane is the optimal compensation hyperplane.Figure 5 shows the compensation results of the calibration and test flight data by using the optimal compensation hyperplane.
In Figure 5, the dash line is the uncompensated data, which is the original data processed by a filter.The uncompensated data corresponds to the left vertical axis.The solid line is the compensated data, which corresponds to the right vertical axis.5 shows that the method based on PCA can obtain good compensation results.Compared with the ridge regression results using the optimal ridge coefficient shown in Figure 6, the method based on PCA has almost the same compensation performance.
In Figure 6, the dash line is the uncompensated data, which corresponds to the left vertical axis.The solid line is the compensated data, which corresponds to the right vertical axis.
The standard deviation and IR are used to assess the results of the different methods.The quantitative comparisons of two methods are presented in Table 2.
Table 2 shows that both of these compensation methods can compensate OPM data well.The performance of the PCA-based method is almost the same or slightly better than the ridge regression method.Moreover, the PCA-based method is only needed to choose the number of PCs entering the model from 1 to 18, which is easier to realize in practice.

Conclusion
A novel PCA-based aeromagnetic compensation method is proposed in this paper.Through extracting the PCs of the feature matrix, multicollinearity in the model is

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Journal of Sensors eliminated.The method enables effective removal of the platform-generated fields of the helicopter.The validity of the proposed method is experimentally demonstrated.The experimental results show that this method has the same or slightly better performance than the ridge regression method.Meanwhile, the method has no need to calculate an optimal ridge coefficient, and this simplifies the aeromagnetic compensation procedure.

Figure 1 :
Figure 1: Structure of the algorithm.

Figure 2 :Figure 3 :
Figure 2: Helicopter and flight path: (a) helicopter mounted with straight rod and (b) calibration flight path and test flight path in Google earth.

Figure 5 :Figure 6 :
Figure 5: Aeromagnetic compensation by PCA-based method: (a) calibration flight data and (b) test flight data.

Figure 4 :
Figure 4: IR-PCs results: (a) calibration flight data and (b) test flight data.

Table 2 :
Standard deviation and IR of different methods.