Attitude angles are important for geomagnetic vector observations. Traditionally, the determination of the azimuth of the fluxgate magnetometer by the staff is made in conjunction with the fluxgate theodolite according to the geographical orientation (Jankowski and Sucksdorff 1996). The tilt of the instrument above the abutment is recorded in real time using a high-precision tiltmeter or a suspended fluxgate magnetometer to circumvent the effect of the tilt. However, in practice, tilt effects are often ignored for observatories in non-permafrost environments. Unattended platforms, such as the SeaFloor ElectroMagnetic Station, use direct attitude measurements by means of tiltmeters and fiber optic gyroscopes (Toh et al. 2006). Researchers have also tried to correct the attitude between two fluxgate magnetometers using genetic algorithms (Liu et al. 2019). A genetic algorithm is a computational model that simulates the process of natural selection and genetic evolution observed in biological evolution; this is a well-established method for finding the global optimal solution of an objective function (Holland 1992; Weile and Michielssen 1997). Earlier studies have used genetic algorithms to calibrate the orthogonality of fluxgate magnetometer probes (Jiao et al. 2011). Three-component fluxgate magnetometer data naturally contain attitude information, and ideally the difference in the data between the two sets of instruments only arises from the difference in the attitude angles. Accordingly, the genetic algorithm calculates the attitude relationship between two fluxgate magnetometers and has natural advantages such as high measurement accuracy, no interference with the probe, and a minimal use of peripheral instruments.
The objective function of the genetic algorithm used in this paper is the difference between the value of the tested instrument data normalized to the standard instrument coordinate system and the value of the standard instrument data, as shown in Equation (1).
![](data:image/png;base64,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)
In Equation (1), the subscripts standard and test represent the data in the standard coordinate system and in the tested coordinate system, respectively, and the superscript ' represents data converted from another coordinate system. The rotation matrix T is obtained by multiplying the rotation matrices represented by the tilt angles Roll, Pitch, and the heading angle Yaw in sequence during the conversion from the standard coordinate system to the tested coordinate system, as shown in Equation (2).
![](data:image/png;base64,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)
This order cannot be changed because the attitude angles correspond to the rotation angles only when the matrix is multiplied in this order.
The presence of deviations in the attitude angles between the two fluxgate magnetometers results in a geomagnetic vector difference in one direction, reflecting a morphological change in the other direction due to projection (Wang et al. 2017). Taking the deviation of the heading angle Yaw as an example, as in Figure 1, the difference dH of the H component reflects the morphological variation of the D component.
Even for the same probe, the characterization factor, that is, the scale factor, used to convert the voltage to the magnetic induction intensity during the instrument commissioning process is not the same. Therefore, the scale factor matrix also needs to be added to the objective function, as in Equation (3).
![](data:image/png;base64,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)
Here, the matrix Sf is the scale factor matrix, which is a diagonal array with its diagonal elements being the scale factors of the corresponding components. The scale factor naturally exists and acts on the tested instrument data first, as in Equation (4); therefore, the attitude rotation matrix and the scale factor matrix acting on the tested instrument data are not interchangeable.
![](data:image/png;base64,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)
Note that the attitude angle and scale factor parameters are for the instrument being tested relative to the standard instrument. The positive direction of the angle is a counterclockwise rotation around the rotation axis, and each scale factor consists of the tested instrument data divided by the standard instrument data.
The presence of the scale factor between the two instruments results in a geomagnetic vector difference in one direction, reflecting a morphological change in this same direction. As shown in Figure 1, the D component difference, dD, exhibits its own morphological variation, that is, it exhibits the morphological variation of the D component.
In the literature (Liu et al. 2019), after selecting the appropriate genetic algorithm parameters, the tested instrument data are first obtained by an angular rotation of the standard instrument data and then a genetic algorithm is used to calculate the attitude angle between the data with an accuracy of up to 10−4 nT(the maximum absolute value of the difference).We can make slight improvements by running each attitude angle calculation 10 times; then, after excluding results that are more than one standard deviation from the mean, the mean value can be used as the attitude angle solution to further constrain the convergence of the genetic algorithm solution. In this way, the maximum absolute difference between the tested instrument data and the standard instrument data after the attitude angle correction can reach 10−5 nT.
Actual observations are often not so ideal. During the period of June 2–July 15, 2021, we conducted related experiments at the LIJ. Two sets of fluxgate magnetometers of the same type were used to make comparative observations. The standard instrument and the tested instrument were spaced 7-m apart, and both were placed on a stone pier in the variation room, with a deviation of 30° in the heading angle between the two. The 7 days with the smallest standard deviation for the results calculated by the genetic algorithm were taken, and the results are shown in Table 1.
Table 1. Calculated results of the comparative attitude angle observation experiment.
No.
|
Roll (°)
|
Pitch (°)
|
Yaw (°)
|
1
|
-2.211
|
-0.437
|
30.000
|
2
|
-0.831
|
-4.140
|
30.000
|
3
|
-1.978
|
2.434
|
30.000
|
4
|
-0.984
|
0.791
|
29.997
|
5
|
-0.512
|
1.158
|
29.943
|
6
|
-1.192
|
1.060
|
29.942
|
7
|
-1.050
|
1.545
|
29.907
|
Average
|
-1.251
|
0.345
|
29.970
|
Standard deviation
|
0.572
|
1.997
|
0.035
|
The calculated heading angle of 29.970° is very close to 30° with a small standard deviation of 0.035°, which indicates that the method used here is able to calculate large existing angles between two fluxgate magnetometers with high accuracy. However, smaller angles, where the standard deviation of the calculated angles over multiple days is greater than the mean value, need to be considered separately.
In the comparative geomagnetic vector observations, the static attitudes of the two fluxgate magnetometers often differ to some extent. Furthermore, the vector difference between the tested instrument data and the standard instrument data after the attitude angle correction is stabilized within approximately 0.5 nT. This requires that the standard deviation of the calculated angle be less than 0.057° (e.g., for a residual magnetic field of 100 nT). However, this requirement is not always satisfied as a result of the quality of the data or disturbances in the observational environment. Therefore, when designing the correction process, we chose, as the criterion to judge the correct calculation of the attitude angles, the uncertainty of all three angles of the multi-day calculation results to be less than 0.1° or the uncertainty of the calculation results of significantly large attitude angles (greater than 1°) to be less than 0.057°.
As for the scale factor, the calculated uncertainty (expressed as the standard deviation of the multi-day scale factors) is small. It is approximately 0.002 for observatories with a good observational environment, resulting in an uncertainty of less than 0.1 nT for the magnetic field. In this paper, after determining the relative attitude angles of the two instruments, we calculate the scale factor for each day in 3 consecutive months and perform linear regression to obtain the base scale factor (intercept) and the long-term time drift (slope). Instrumental scientists and engineers assume that the long-term time drift of a fluxgate magnetometer is linear (Gordon and Brown 1972; Esper 2020); such behavior is characterized by tiny fluctuations in the short-term observations and non-negligible and linear variations in the long-term observations. If the long-term time drifts of the two instruments are different, there will be a linear change in the scale factor between the instruments over time. This is a relative relationship and does not specify whether the drift comes from the tested instrument or the standard instrument or both. However, for the correction of the long-term observational agreement, it is sufficient to assume that the drift comes from the tested instrument.
Temperature has an important effect on fluxgate magnetometers (Primdahl 1979). New fluxgate magnetometers have been able to achieve a thermal drift of less than 0.1 nT/°C in the laboratory (Korepanov 2012). However, the effect of temperature on fluxgate magnetometers is still very important and non-negligible in long-term observations. Even though the temperature difference between two sets of instruments in the same variation room is nearly constant (the mean value of the temperature difference between the two sets of instruments at LIJ from January 1, 2020, to March 31, 2020, was 0.9672 °C, and the standard deviation was 0.0863 °C), the measurement difference caused by the fixed temperature difference is not constant, which means that the temperature change affects the measurement difference between the two sets of instruments. The top section of Figure 2 shows the Z component difference, dZ, and temperature variation of the two sets of instrumental data from January to March 2020 at LIJ after the attitude angle, scale factor, and long-term time drift corrections. The dZ pattern, showing a decrease followed by an increase, is very similar to the temperature variation. This relationship is approximately linear, as seen in the scatter plot of dZ versus temperature (middle section of Figure 2). The temperature, rather than the temperature difference, has a linear effect on the vector difference between the two sets of instruments. This can be interpreted as the difference in the temperature coefficients between the two sets of instruments leading to differences in the measurements at different temperature points (the red line at the bottom of Figure 2), even though the temperature difference is always constant (the blue line at the bottom of Figure 2).
The morphological differences and the data disagreement in the comparative geomagnetic vector observations are primarily due to the attitude angle, scale factor, long-term drift, and relative temperature coefficients. The calculations of the attitude angle and the scale factor are based on the genetic algorithm. The long-term time drift and relative temperature coefficients are then obtained via linear regression. The parameter calculation and correction flow used in this paper are shown in Figure 3.