## Abstract

By detecting the severe meteorological situations on flight route, airborne weather radar (WXR) can ensure the safety of the aircraft and on-board personnel. Among these critical weather conditions, atmospheric turbulence is one of the main factors that affect flight safety. Atmospheric turbulence detection method that the current WXR adopts mainly is the pulse pair processing (PPP) method, which estimates Doppler spectrum width of weather target echo and compares it with a threshold to determine whether this weather target is turbulence or not. PPP method is simple and easy to implement, but the performance of this method under the condition of low signal-to-noise ratio (SNR) is poor. In this paper, we propose a new turbulence detection method based on the principal component analysis (PCA) approach. This new method uses PCA approach to preprocess the weather target echo and divides it into two parts: the principal component part as signal and the rest part as noise, so as to realize the de-noising function of PCA approach, and it is then combined with PPP method to estimate the spectrum width. Due to the good de-noising performance of PCA approach, this new method improves the detection performance of traditional PPP method especially under the condition of low SNR.

## Keywords

Airborne weather radar Atmospheric turbulence detection Pulse pair processing Principal component analysis De-noising## 1 Introduction

Airborne weather radar (WXR), as a subsystem of the aircraft environment surveillance system (AESS), can help to ensure the safety of aircraft and on-board personnel under severe weather conditions by detecting the weather conditions within a certain fan-shaped area in front of the aircraft. Among these severe weather conditions, atmospheric turbulence is one of the important factors affecting aircraft flight safety. Because of its random fluctuations within a certain period of time and space, severe turbulence will not only make the aircraft jolt but also affect the safety of the on-board personnel.

The detection of meteorological targets by WXR is accomplished through a series of processing of radar echoes reflected from meteorological targets. The three spectrum moments of meteorological target echoes are closely related to the types of meteorological targets [1, 2, 3]: the magnitude of echo power or the zero moment of the Doppler spectrum can be used to determine the intensity of rainfall on the flight path; by calculating the rate of change of the average wind speed with distance, we can determine the degree of danger of wind shear; by comparing the spectrum width of the echo and the spectrum width threshold of turbulence, we can determine whether the front of the aircraft has turbulent area causing bumps. Therefore, the detection of atmospheric turbulence targets is essentially an estimation of the spectrum width of radar echoes.

At present, the methods of estimating spectrum width used in actual airborne weather radars are mainly the pulse pair processing (PPP) method and the fast Fourier transform (FFT) method [4, 5, 6]. The PPP method estimates the Doppler spectrum width by calculating the correlation function of the adjacent pulse echoes; the FFT method calculates the power spectrum of the echoes by the fast Fourier transform, and estimates Doppler spectrum width according to the second order moment of power spectrum. These two methods are simple to compute and provide adequate performance at high signal-to-noise ratio (SNR), but the performance is poor when SNR is low. Therefore, this paper proposes a new spectrum width estimation method using principal component analysis (PCA) approach for noise reduction. Through the principal component analysis of radar echoes, the principal part is regarded as the signal subspace, others as the noise subspace so as to achieve the purpose of removing noise and improving the spectrum width estimation accuracy.

## 2 Radar echoes simulation

First, we need to simulate the radar echoes of the meteorological target. Compared with the actual radar echoes, the parameters (average frequency and spectrum width) of the simulation signal are adjustable, making it easy to study the feasibility of new methods.

The meteorological targets are distributed targets, which include many scattering points in each radar resolution unit. The echo of each unit is the superposition of the echoes of all these scattering points, so it is generally considered that the echo signal of the meteorological target is Gaussian signal, in which amplitude and power spectrum obey the Gaussian distribution [7, 8].

*G*

_{n}is the discrete power spectrum coefficient corresponding to

*f*

_{n}, \( \bar{f} \) is the average Doppler frequency, and

*σ*

_{f}is the Doppler spectrum width. The echoes generally contain Gaussian white noise, thus meteorological target echo power spectrum sequence superimposed with noise can be expressed as

*x*

_{n}is random number following the uniform distribution (0, 1), and

*K*is the parameter to control the SNR:

*P*

_{N}is the noise power, generally set to 1. SNR is the signal-to-noise ratio in dB, and

*N*is the length of the spectrum sequence.

*I*,

*Q*, the phase spectrum of the signal must also be considered, which is irrelevant and uniformly distributed over (−

*π*,

*π*). Decomposing the power spectrum sequence into real parts and imaginary parts to obtain a complex spectrum:

*y*

_{n}is random number following the uniform distribution (0, 1).

At this point, an airborne weather radar echoes with superimposed white noise obeying a Gaussian spectrum distribution can be generated.

## 3 Traditional PPP method

The traditional PPP method was first proposed by Rummler [9] and has been widely used in actual WXR systems. It estimates the target Doppler mean velocity and spectrum width by calculating the correlation function of the adjacent echo signals [10].

*N*, then according to the definition of autocorrelation function,

According to the Doppler effect, the corresponding values of the average velocity and velocity spectrum width can be obtained.

## 4 PPP based on PCA de-noising

The method proposed in this paper is first using the PCA approach to de-noise and then combining with PPP method to estimate the spectrum width. The initial application of the PCA is dimension reduction, that is, through orthogonal transformation, a set of high-dimensional data with correlation is converted into a set of low-dimensional data composed of principal components that have no correlation. In this paper, we use principal component analysis to decompose the autocorrelation matrix of the echo data into principal components and minor components. The principal components are considered as signal subspaces and the rest are noise subspaces. The purpose of noise reduction is achieved by retaining the signal subspace only [11]; Benchebha et al. [12, 13].

*s*represents the signal,

*n*represents the additive noise, and

*N*represents the number of pulses, i.e., the sequence length. Then the autocorrelation matrix of the echo data can be expressed as a sum of the autocorrelation matrices related to the signal part and the autocorrelation matrices related to the noise part. In practice, the autocorrelation matrix is usually unknown but can be estimated from the sample data. If the matrix size is assumed as

*N*×

*N*, then

*r*is the number of realization of the process, which means

**Z**(

*r*) =

^{def}

**Z**.

*λ*

_{i}denotes the eigenvalues and is assumed to be in descending order (

*λ*

_{1}≥

*λ*

_{2}≥ … ≥

*λ*

_{N}),

**V**

_{i}is an eigenvector corresponding to the eigenvalue

*λ*

_{i}.

*p*major components, then

Since we only keep the signal part and remove the noise, the performance of the spectrum width estimation will be enhanced. Now, any spectrum moment estimator can use this autocorrelation matrix to estimate the spectrum width. In this paper, the PPP method is combined with PCA to estimate the spectrum width.

## 5 Simulation and analysis

In this section, we will focus on comparing the estimated performance of PPP method before and after PCA de-noising. We introduce the relative error as a measure criterion, the smaller the error, the better the accuracy of the estimation.

In order not to be affected by the influence of radar pulse repetition frequency, the parameters in this simulation adopt normalized values. First, we set the power as 1, average Doppler frequency as 0.1, Doppler spectrum width within 7 range cells as [0.07, 0.075, 0.08, 0.085, 0.09, 0.095, and 0.1]. According to Eqs. (1)–(5), radar echo sequences with different spectrum widths are generated.

When the sequence length *N* is 256 and the SNR is 20 dB, the simulation results are shown in the following figures:

*σ*

_{real}represents the true normalized spectral width of 7 range cells,

*σ*

_{ppp}is the spectral width value obtained by the PPP method, and

*σ*

_{pca}is the spectral width value estimated by the proposed PCA method. It can be generally seen from the result of Fig. 1 that the estimated value obtained by the proposed method is closer to the true value. Figure 2 shows the relative error between the real value and the estimated value of the spectrum width obtained by the two estimation methods.

*E*

_{ppp}is the relative error of the PPP method and

*E*

_{pca}is the relative error of the PCA method. It can be clearly seen that the proposed method has a smaller error, that is the higher estimation accuracy.

When the sequence length *N* is 256 and the SNR is 10 dB, the simulation results are shown in the following figures:

*σ*

_{real}represents the true normalized spectral width of 7 range cells,

*σ*

_{ppp}is the spectral width value obtained by the PPP method, and

*σ*

_{pca}is the spectral width value estimated by the proposed PCA method. From the results in Fig. 3, it can be seen that although the two estimation methods are very different from the real values, the proposed method estimates are closer to the real spectral width. Figure 4 shows the relative error between the real value and the estimated value of the spectrum width obtained by the two estimation methods.

*E*

_{ppp}is the relative error of the PPP method and

*E*

_{pca}is the relative error of the PCA method. Although the estimation errors of the two methods are both high in the case of low SNR, the proposed method can still improve the estimation error of the PPP method and the estimation result is more accurate.

When the sequence length *N* is 128 and the SNR is 20 dB, the simulation results are shown in the following figures:

The relative error of spectral width estimation (%)

Range cell |
SNR = 20 dB |
SNR = 10 dB |
SNR = 20 dB | |||
---|---|---|---|---|---|---|

PPP | PCA | PPP | PCA | PPP | PCA | |

1 | 4.55 | 1.45 | 33.77 | 31.73 | 6.24 | 2.48 |

2 | 3.95 | 1.45 | 29.07 | 27.48 | 6.39 | 2.54 |

3 | 3.51 | 1.69 | 26.29 | 23.07 | 5.55 | 2.42 |

4 | 3.66 | 1.86 | 21.11 | 19.90 | 5.49 | 2.68 |

5 | 3.10 | 2.53 | 18.07 | 16.65 | 5.48 | 3.12 |

6 | 4.13 | 3.39 | 16.15 | 14.16 | 5.53 | 3.83 |

7 | 4.35 | 3.83 | 13.11 | 11.68 | 6.01 | 4.47 |

Average | 3.89 | 2.31 | 22.51 | 20.67 | 5.81 | 3.08 |

## 6 Conclusion

To improve the performance of spectral width estimation under low SNR in traditional PPP method, this paper proposes a spectral width estimation method using PCA approach to de-noise. Through the principal component analysis of autocorrelation function matrix of radar echo data, the autocorrelation function matrix related to the signal alone is obtained, and then the spectrum width estimation is performed with the PPP method.

The simulation results show that the estimated error of the proposed method is smaller than that of the PPP method regardless of whether the SNR is high or low. And when the data length is short, the proposed method is still valid. The analysis and simulations show that the proposed method can improve the accuracy of traditional PPP method’s spectral width estimation, and thus can be expected to accurately detect atmospheric turbulence in practical applications.

Although the proposed method can improve the estimation performance under low SNR, the estimation error is still high. Next we will use this new method to process the actual radar data to find out whether some parameters need to be adjusted.

## Notes

### Acknowledgments

This paper is sponsored by National Program on Key Basic Research Project (2014CB744903), National Natural Science Foundation of China (61673270), Shanghai Pujiang Program (16PJD028), Shanghai Industrial Strengthening Project (GYQJ-2017-5-08), Shanghai Science and Technology Committee Research Project (17DZ1204304), and Shanghai Engineering Research Center of Civil Aircraft Flight Testing.

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