Blind deconvolution algorithm for gravity anomaly distortion correction

ZHAO Li-ye, LI Hong-sheng

(Key Laboratory of Micro Inertial instrument and Advanced Navigation technology, Ministry of Education, School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China)

Blind deconvolution algorithm for gravity anomaly distortion correction

ZHAO Li-ye, LI Hong-sheng

(Key Laboratory of Micro Inertial instrument and Advanced Navigation technology, Ministry of Education, School of Instrument Science and Engineering, Southeast University, Nanjing 210096, China)

Strong damping and large time constant are the common characteristics of marine gravity meter, which can suppress the interference of vertical acceleration, but can also lead to distortion of the lowfrequent gravity anomaly signals, such as amplitude attenuation and phase lag. In order to suppress serious background noises and get high-precision gravity information from the measured signals of the precise gravimeter, a new method of single-channel Bussgang algorithm is proposed based on the principle of the gravimeter and the distortion of the measured signals, and is applied to the correction of gravity anomaly distortion. In the signal processing procedure, the deconvolution filter is simplified as a FIR model, and then the single-channel Bussgang deconvolution algorithm - which uses the constant modulus algorithm (CMA) in updating equations - is used to estimate the deconvolution filter. Finally, the measured gravity signal is used for comparing the proposed method with Kalman inverse filter. Emulations and experiments indicate that the proposed single-channel Bussgang algorithm has better performance than that of Kalman inverse filter in alleviating the distortion of the gravity anomaly signal. The distortion correction standard deviations of the proposed method and the inverse Kalman filter are 0.328×10-5m/s2and 1.838×10-5m/s, respectively.

gravimeter, gravity anomaly; Bussgang deconvolution algorithm; Kalman inverse filter; distortion correction

Gravity is one of the extremely important parameters for numerical calculation and control in gravity/inertial navigation system, so the accuracy of gravity determines the precision of the navigation system. In such system, a referenced ellipsoid model is commonly used to describe the gravity field. For the whole shape of the Earth surface, the referenced ellipsoid model is an excellent approximation, but for some local region, such as marine gravitywith complex geological situation, the model would introduces errors. Along with the development of the inertial instrument performance, the errors of inertial components are no longer the most crucial factors degrading the precision of the gravity/inertial navigation system. Gravity anomaly – the difference between the real gravity and the measured gravity – has been regarded as the greatest error resource in high precision gravity/ inertial navigation system[1-4]. So far, the improvement of system precision relies on high accuracy of gravity information. Thus, real-time measurement and error-correction method of gravity anomaly are essential to extend the system operation time in gravity/inertial navigation system with high precision.

Only few papers discussed the problem of the gravity anomaly distortion correction. A posterior correction algorithm was proposed in document [5]. Document [6] proposed a Kalman inverse filter scheme to correct the gravity anomaly distortion and testified the feasibility of the algorithm through simulations. However, the ideal situation was supposed in the simulation of the scheme, such as ignoring the measurement noise and not considering the influence of the high frequent disturbance in the original gravity anomaly signal, etc. Thus, the performance under severe environmental noises and disturbances should be discussed through further researches. This paper proposed a single-channel Bussgang algorithm for gravity anomaly distortion correction. Simulations and experiments indicate that the proposed method achieves better performance than that of Kalman inverse filter method.

1 Distortion principle of gravimeter signals

The most important part of sea gravimeters is a zerolength spring, and the principle of the sensor is shown in Fig.1.

Fig.1 Zero-length spring gravimeters

Variations in the vertical acceleration (gravity plus ship acceleration) cause the mass removed from its original position. The amount of displacement indicates the amplitude of acceleration, and it is recorded as the differential voltage.

Since the gravity measurement signal is inevitably disturbed by vertical acceleration (sometimes, it is more than 200 gal), so an attenuation in excess of 105is required when the gravity anomaly precision is 1 mGal

Usually, in order to constrain the spring and mass oscillation and suppress high-frequency disturbances to an acceptable level, the sensor part of the sea gravimeter is placed in the silicone oil with strong damping. The silicone oil damping is equivalent to an unknown lowpass filter with a long time constant. Therefore, the amplitude reduction and phase lag is introduced in the output of the sea gravimeter by the unknown low-pass filter.

The spring-mass system shown in Fig.1 is a typical second-order model, the motion equation of the system can be actually described by the following second-order equation

where x(t) is the displacement of the mass from its original position, k is the stiffness of the zero-length spring, a(t) is the input acceleration of the mass in vertical direction, including marine gravity anomaly signal and vertical interference acceleration, and λ is the damping coefficient.

Because it is over-damped system, the second-order term in Eq. (1) can be neglected and the motion equation could be simply expressed as follow:

The Laplace transform of the Eq. (2) is

where G1(s) is the transfer function of the spring-mass system, T1=λ/k, K1=m/k.

2 Proposed single-channel Bussgang algorithm

The blind deconvolution is a signal processing method which is widely used in several applications, e.g. blind image deconvolution[7]and mobile communication[8]. Fig.2 shows the general blind deconvolution schematic, where the source signal s(t) is filtered by an unknown transmission channel, which is denoted by ak.

Fig.2 Blind deconvolution system

The measured signal x(t) can be expressed as

The measurement signal x(t) contains mixtures of the source signal elements at multiple lags with additive noise n(t), which can be neglected to simplify the analysis. And the mixing process is deconvolved by a linear filtering operation, which is denoted by deconvolution filter wk.

For simplicity, the deconvolution filter can be approximately expressed as the FIR model shown in Fig.3.

Fig.3 Structure of the inverse filter

The output of the deconvolution filter wi(t) is

The blind deconvolution is solved when[9]

In such situations, the original source signal is recovered by the blind deconvolution system. In order to achieve the purpose of blind deconvolution, the method of singlechannel Bussgang is applied to the deconvolution filter wkto achieve the optimal deconvolution, then the source signal s(t) can be the accurately recovered.

It is well-known that many Bussgang-type adaptive algorithms have good performances in single-channel deconvolution, and the diagram of the algorithm can be shown as the following Fig.4.

Fig.4 Bussgang algorithm

The following equations can be used to update the deconvolution filter W=(w-L, w-L+1, …, w0, wL).

where g(?) is a nonlinear function, and the performance of the deconvolution based on Bussgang algorithm depends on the amplitude distribution of the source sequence and the choice of nonlinear functions and the update of the deconvolution filter W.

Different nonlinear functions g(?) can obtain different algorithms which include the Sato, the Godard and constant modulus algorithms(CMA)[10-11]. Constant modulus algorithm is consistently widely used in the Bussgang of blind deconvolution algorithms for the signal processing, and the nonlinear function is

The parameter2γis known as a constant, which is normally defined as the statistics of source signal.

The cost function of CMA is constructed by the higher order statistical characteristics of the transmission signal and can be expressed as following:

The CMA can be considered as a stochastic-gradient procedure for minimizing the cost function, and the deconvolution update of the CMA is

where y(t) is the measurement signal, the parameter μ controls the convergence performance of the algorithm and is normally chosen as a positive number.

3 Kalman inverse filter state equations and measurement equations of gravimeter for distortion correction

In order to use Kalman inverse filter to alleviate the distortion of the gravity anomaly signal, the system state equation and measurement equation of the gravimeter must be firstly established.

Suppose the discrete state equation of gravimeter is

and the measurement equation is

According to the analysis in document [6], the gravity anomaly signal could be regarded as the output of linear system stimulated by white noise, so the discrete state equation of anomaly signal is

Combining with Eq. (12)-(15), the matrix equation can be concluded as following:

According to Eq. (16) and Eq. (17), the state variable X2(k)can be estimated by using Kalman algorithm[12], then the estimation of gravity anomaly signal S(k)can be obtained by Eq. (17).

4 Simulations and experiments of gravity anomaly distortion correction

The sinusoidal signal with high frequency noise is assumed as the output signal of gravimeter, which is shown in Fig.5. The processed result of Kalman inverse filter and the proposed single-channel Bussgang algorithm is shown in Fig.6, Fig.7. In Fig.6, the solid line represents the ideal gravity anomaly and the dot line stands for the estimation result of Kalman inverse filter. In Fig.7, the solid line represents the ideal gravity anomaly and the dot line stands for the estimation result of the proposed singlechannel Bussgang algorithm.

By the same way, a linear signal with high frequency noise is also assumed as the output signal of gravimeter, which is shown in Fig.8. The processed result of Kalman inverse filter and the proposed single-channel Bussgang algorithm is shown in Fig.9 and Fig.10. In Fig.9, the solid line represents the ideal gravity anomaly, and the dot line stands for the estimation result of Kalman inverse filter. In Fig.10, the solid line represents the ideal gravity anomaly, and the dot line stands for the estimation result of the proposed single-channel Bussgang algorithm.

The standard deviation of the two types of signals corrected by Kalman inverse filter and the proposed method are shown in Tab.1.

Based on Fig.6, Fig.7, Fig.9, Fig.10 and Tab.1, we can make the conclusion that the proposed single-channel Bussgang algorithm can achieve better performance of gravity anomaly distortion correction.

Tab.1 Standard deviation of corrected signal by Kalman inverse filter and the proposed method

Fig.5 Sinusoidal output signal of the gravimeter

Fig.6 Estimation result of Kalman inverse filter compared with the real signal

Fig.7 Estimation result of the proposed single-channel Bussgang algorithm compared with the real signal

Fig.8 Linear output signal of the gravimeter

Fig.9 Estimation result of Kalman inverse filter compared with the real signal

Fig.10 Estimation result of the proposed single-channel Bussgang algorithm compared with the real signal

In order to confirm the performance of the proposed single-channel Bussgang algorithm, both Kalman inverse filter and the proposed single-channel Bussgang algorithm are used to deal with the measured marine gravitydata which is shown as the solid line in Fig.11 and Fig.12. In Fig.11, the result of Kalman inverse filter is indicated by dot line, compared with the ideal signal represented by solid line. And the result of the proposed single-channel Bussgang algorithm is also indicated by dot line, as shown in Fig.12. And the standard deviation of the estimation result of Kalman inverse filter with the real signal is 1.838×10-5m/s2, while the standard deviation of the estimation result of the proposed single-channel Bussgang algorithm with the real signal is 0.328×10-5m/s2. Therefore, we conclude that the performance of the proposed single-channel Bussgang algorithm is greatly improved compared with that of Kalman inverse filter.

Fig.11 Estimation result of Kalman inverse filter compared with the real signal

Fig.12 Estimation result of the proposed single-channel Bussgang algorithm compared with the real signal

5 Conclusion

Amplitude attenuation and phase lag are main kinds of distortions in marine gravimeter with strong damping and large time constant. In order to overcome the drawbacks and improve the precision of gravity anomaly measurement, the method of distortion correction should be applied. Based on the Bussgang deconvolution algorithm, a new method of single-channel Bussgang algorithm is proposed and applied to the correction of gravity anomaly distortion. Emulations and experiments indicate that the performance of the proposed algorithm is better than that of Kalman inverse filter method. The result of this research is valuable for gravity/inertial navigation system by improving the precision of gravity anomaly information.

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一種用于重力測量信號畸變校正的盲反卷積算法

趙立業，李宏生

( 1. 微慣性儀表與先進導航技術教育部重點實驗室，南京 210096 ；2. 東南大學 儀器科學與工程學院，南京210096 )

強阻尼和大時間常數是海洋重力儀的共同特性，它們在抑制垂直方向上干擾加速度的同時也造成了重力儀測量信號的幅值衰減和相位滯后。為了校正重力測量信號的畸變，獲得高精度的重力異常信號，在分析重力儀和重力測量信號畸變原理的基礎上，提出了一種單通道Bussgang算法，并應用于重力測量信號的畸變校正。該方法首先將盲反卷積濾波器近似簡化為FIR模型，然后采用常數恒模算法對Bussgang均衡器權系數進行更新，并在此基礎上對反卷積濾波器進行了估計，最后基于實測重力信號將該方法與Kalman逆濾波進行了試驗對比。理論分析和試驗結果表明，該重力信號畸變校正方法能有效地消除重力測量信號的畸變，且校正效果明顯優于Kalman逆濾波，其校正后信號相對于原始信號的標準差為0.328×10-5m/s2，而Kalman逆濾波校正標準差為1.838×10-5m/s2。

重力儀；重力異常；Bussgang 反卷積算法；Kalman逆濾波；畸變校正

U666.1

A

1005-6734(2015)02-0196-05

2014-11-15；

2015-03-11

國家自然科學基金資助項目（61101163）；江蘇省自然科學基金資助項目（BK2012739）

趙立業（1977—），男，副教授，博士生導師，從事高精度重力信號處理研究。E-mail：liyezhao@seu.edu.cn

10.13695/j.cnki.12-1222/o3.2015.02.011