## 4 The Proposed Method and Principle
### The Improved Coherent Noise Suppression (ICNS) Method  
The CNS is a practical noise cancelation method. The distance between the two magnetometers is far less than the distance between the helicopter and the target, so it can be considered that the two magnetometers in the helicopter measure the common value of the magnetic field from the target and the different values from the interference field. In addition, it is known that the vibration interference field of the two magnetometers is highly correlated. After the compensation, we obtained the magnetic field data sequences \(y_{1}(n)\) and \(y_{2}(n)\). The \(y_{1}(n)\) involves the signal field \(s(n)\) and interference field \(d(n)\), while \(y_{2}(n)\) involves the signal field \(s(n)\) and stronger interference field \(d_{2}(n)\). The \(n\) denotes the sample index, where \(w_{1}(n)\) and \(w_{2}(n)\) represent the intrinsic noises of OPM-I and OPM-II, respectively. \[ \begin{array}{l}y_{1}(n)=s(n)+d(n)+w_{1}(n)\\ y_{2}(n)=s(n)+d_{2}(n)+w_{2}(n)\end{array} . \tag{9}\]  
Considering that the two magnetometers were subject to different interference levels from helicopters, the interference magnetic field that was measured by the closer magnetometer was larger and contained more interference components. In order to avoid the influence from the incoherent noise of the two magnetometers with the CNS method, it was noticed that the OPM-II had more interference field characteristics. Before the CNS processing, we needed to preprocess the measured data from the two magnetometers. Here, we proposed a spectral subtraction method to reduce the quasi-stationary components from background interference based on a priori signal-to-noise ratio (PSNR) estimation, which was used to respectively process the measurement data from the two magnetometers. As an example, the magnetic field \(y_{1}(n)\) measured by OPM-I can be expressed as follows:  
\[y_{1}(n)=s(n)+d(n)+w_{1}(n). \tag{10}\]  
In the process, this signal was evenly divided into short-time frames and analyzed frame by frame, assuming that its behavior was similar over the duration of all the analyses frames. For a short time frame, \(k\) defines the spectral component of the noisy signal as \(Y_{1}(k,\,\omega)\), the signal as \(S(k,\,\omega)\), and the interference and noise as \(D(k,\,\omega)\). The interference reduction process was used for each short-time frequency spectrum, \(Y_{1}(k,\,\omega)\), with a  
Figure 6: The model used for the adaptive interference cancelation method. spectral gain, \(G(k,\omega)\), which is a function of two parameters in the signal-to-noise ratio with a priori, \(SNR_{prio}(k,\omega)\), and a posteriori, \(SNR_{post}(k,\omega)\), as shown in Equations (11)-(13). \[SNR_{prio}(k,\omega)=\frac{\left|Y_{1}(k,\omega)\right|^{2}}{E\Big{\{}\left|D(k,\omega)\right|^{2}\Big{\}}}. \tag{11}\]  
\[SNR_{post}(k,\omega)=\frac{\left|S(k,\omega)\right|^{2}}{E\Big{\{}\left|D(k, \omega)\right|^{2}\Big{\}}}. \tag{12}\]  
\[G(k,\omega)=\sqrt{\frac{SNR_{prio}(k,\omega)}{SNR_{prio}(k,\omega)+1}}. \tag{13}\]  
The noise power spectral density \(E[\,|\,D(k,\omega)\,|^{2}]\) can be estimated during the periods when the signal from the target is not present, where \(E\) is the expectation operator. The \(SNR_{prio}(k,\,\omega)\) can be obtained through the following recursive relation, where \(\alpha\) is the smoothing coefficient, which is generally a value between 0.8 and 1. \[SNR_{prio}(k,\omega)=\alpha\frac{\left|S(k-1,\omega)\right|^{2}}{E\Big{\{} \left|D(k,\omega)\right|^{2}\Big{\}}}+(1-\alpha)\cdot max\big{[}SNR_{post}(k, \omega)-1,0\big{]}. \tag{14}\]  
Then, \(S(k,\omega)\) is the estimated signal spectrum at the previous frame and is obtained as follows:  
\[S(k,\omega)=G(k,\omega)\cdot Y_{1}(k,\omega). \tag{15}\]  
After the preprocessing, the outputs of OPM-I and OPM-II can be expressed as Equations (16) and (17), where \(s(n)\), \(e_{1}(n)\), and \(e_{2}(n)\) represent the target signal, and the residual interference field in OPM-I and in OPM-II, respectively. \[