A method of locating a magnetic target based on geomagnetic total field is proposed. In the method, a conjugate gradient algorithm is introduced to eliminate the time-varying and uneven spatial distribution of geomagnetic total field. Then a structure of the measuring array of geomagnetic total field is designed. In the measuring array, the array aperture is a primary factor for the conjugate gradient algorithm. To determine an optimal aperture, we analyze the relationship between the array aperture and the localization accuracy. According to the localization theory based on geomagnetic total field, we simulate the process of determining an optimum array aperture. Based on the simulation, we propose the basis and principle of determining the optimum array aperture. To prove it, we use optically pumped magnetometers with different array apertures to carry out the experiments of locating a car in a suburb. Through the experiment, we get the experimental relationship between apertures and location accuracy. And the relationship agrees with the theory. The result shows that the method is feasible to determine the optimum aperture.
It is important to locate targets by magnetic field in geological monitoring, energy and mineral exploration, rescue of plane crash, antisubmarine detection, and medical diagnosis [
The double gradient algorithm comprehensively describes the actual situation of target’s magnetic field gradient, so the algorithm is not affected by the filtering of some signal gradient. The difference is only that the measured value of the decimal point has been moved forward. For the sensors with high precision and high resolution, the accuracy of locating a target will not be affected as long as the mantissa can be measured. However, if the sensor accuracy and resolution are low, the mantissa will not be detected. It means that the dual gradient algorithm cannot detect the remote or small target. An optimum aperture is an equilibrium point of the double gradient algorithm. Base on the optimum aperture, we can filter the interference field to the maximum extent and filter the target field to the minimum extent. It is impossible to eliminate the magnetic field generated by the target completely. It is only required that the desired accuracy of position is within an expected range.
The optimum array aperture is related to sensor precision, resolution, noise of instrument and environment, target magnetic moment, detection position, and detection range. These quantities should be estimated and expected before designing the positioning system, so the calculation method of the optimum aperture can be given according to these estimates.
When the distance between a target and a sensor is more than 2-3 times the scale of the target, the target can be regarded as a magnetic dipole [
Schematic diagram of the location array.
According to the magnetic dipole model in far field, the magnetic field
Substituting
into equation (
Then, the scalar measurement of the
It can be seen that formula (
The time-varying and uneven spatial distribution may have an influence on locating a target. The influence must be eliminated in magnetic field measurement. The measured value of
Through (
In the linear motion of a target, its magnetic moment vector is fixed, which can be regarded as unknown constants. According to formula (
The uncertainty at
In the formula,
In order to improve the positioning accuracy, the function
According to the above principle, the optimal aperture can be determined by the following procedures:
Determinate the instrument and ambient noise Set the location accuracy For specific Find the intersection of Choose the minimum value of
Step 4 can be omitted when a coordinate point or a target in a local area is concerned.
In theory, it is not possible in theory to find the corresponding
As shown in Figure
Schematic diagram and photo of the experimental area.
The aperture of the array is
The curve of the magnetic field changed over time in the experimental area, as shown in Figure
Curve of magnetic field variation over time in experimental area.
Figure
Curves of
Curves of
Figure
Three-dimensional curves of
Figure
Curves of
Curves when
Curves when
Curves when
Curves when
Curves when
Curves when
Curves when
Curves when
Curves when
Therefore, the maximum
Value of
40.00 | |
33.13 | |
26.27 | |
12.55 | |
5.69 | |
-1.18 | |
-14.90 | |
-21.77 | |
-28.63 |
According to the requirements of accuracy, the intersection of
We also performed the simulation with the array of HS-MS-FG3S fluxgate magnetometers made in China. The frequency-domain noise of this magnetometer is 10 pT/Hz1/2@1 Hz per channel. When we measure the total geomagnetic field by it, the total noise of three channels is 17.32 pT/Hz1/2@1 Hz, and the total noise is 20 pT/Hz1/2@1 Hz after superimposing with the white noise of geomagnetic field. When the positioning accuracy is 1 m, the range of
Value of
40.00 | |
33.13 | |
26.27 | |
12.55 | |
5.69 | |
-1.18 | |
-14.90 | |
-21.77 | |
-28.63 |
The intersection of
We set four CS-L cesium optically pumped magnetometers in the suburb as shown in Figure
Experimental data and theoretical curves of
According to formulae (
Results of location experiment.
No. | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 19.52 | 2.12 | -2.22 | 22.82 | 19.42 | 28.08 | 14.21 | 6.53 | 19.42 | 33.49 | 19.92 | 1.55 | 18.76 | 30.61 | 17.22 | 2.09 |
2 | 12.81 | 2.04 | -2.12 | 16.51 | 12.55 | 34.56 | 14.57 | 3.25 | 12.55 | 33.88 | 13.75 | 2.21 | 11.56 | 33.71 | 13.02 | 2.23 |
3 | 6.10 | 7.11 | 4.91 | 2.32 | 5.69 | 34.81 | 7.42 | 3.29 | 5.69 | 32.09 | 5.76 | 0.10 | 4.36 | 33.19 | 5.51 | 1.64 |
4 | -0.61 | 8.48 | -0.09 | 0.81 | -1.18 | 19.70 | -3.85 | 12.61 | -1.18 | 29.61 | -1.97 | 2.53 | -2.84 | 23.90 | -4.36 | 8.26 |
5 | -7.32 | 2.00 | -2.69 | 8.48 | -8.04 | 23.52 | -9.91 | 8.70 | -8.04 | 31.56 | -8.10 | 0.47 | -10.04 | 35.51 | -8.93 | 3.66 |
6 | -14.03 | 16.37 | -2.58 | 13.56 | -14.90 | 24.68 | -18.03 | 7.98 | -14.90 | 27.86 | -16.03 | 4.31 | -17.24 | 35.39 | -17.27 | 3.37 |
7 | -20.74 | 2.84 | -7.21 | 14.91 | -21.77 | 40.00 | -20.80 | 8.04 | -21.77 | 39.06 | -20.49 | 7.16 | -24.44 | 21.01 | -26.17 | 11.15 |
8 | -27.45 | 2.00 | -2.85 | 25.60 | -28.63 | 20.00 | 7.22 | 37.82 | -28.63 | 33.16 | -29.01 | 1.20 | -31.64 | 28.80 | -36.01 | 5.43 |
9 | -34.16 | 2.00 | -2.70 | 32.25 | -35.50 | 30.82 | -36.77 | 1.75 | -35.50 | 33.80 | -32.79 | 3.25 | -38.84 | 31.20 | -41.00 | 2.31 |
15.25 | 9.99 | 2.53 | 4.46 |
In Table
The method of constructing an array of total field magnetometers to locate a target is proposed. This method is based on a magnetic dipole model of far field. We filtered out the time-varying and uneven spatial distribution by the conjugate gradient algorithm. Then the method of determining an array aperture is proposed. We simulate the location algorithm and the process of determining an array aperture. The location experiment is carried out in the suburb. We choose 1 m, 2 m, 4 m, and 6 m, respectively, as the aperture. The experimental data is consistent with the theoretical curve. In the location algorithm, interactive linear and general optimization solver LINGO is used to solve the nonlinear equations. The location accuracy of four apertures is calculated, respectively, which basically agrees with the simulation. In the experiment, the CS-L cesium optically pumped magnetometers are used. The CS-L cesium optically pumped magnetometers have a high resolution. Also, the measurement is not affected by the temperature. The attitude of the magnetometers is not rigidly calibrated. In the location experiment, the diagonal of the car is 4.64 m and the magnetic moment is about
Results of location experiment.
When
When
When
When
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
The measurement values of magnetometers in the location experiment. The data are the original experimental data which are measured by the CS-L array in Figure