Journal of Sensors

Volume 2018, Article ID 5847081, 14 pages

https://doi.org/10.1155/2018/5847081

## A Novel Weak Signal Detection Method of Electromagnetic LWD Based on a Duffing Oscillator

^{1}National Demonstration Center for Experimental Electrical & Electronical Education, Yangtze University, Jingzhou, Hubei 434023, China^{2}Laboratory for Marine Geology, Qingdao National Laboratory for Marine Science and Technology, Qingdao 266061, China^{3}Faculty of Computing, Information Systems and Mathematics, Institute of Finance Management, P.O. Box 3918, Dar es Salaam, Tanzania^{4}Key Laboratory of Marine Sedimentology and Environmental Geology, First Institute of Oceanography, SOA, Qingdao 266061, China

Correspondence should be addressed to Qingqing Fu; nc.ude.ueztgnay@qqfupj

Received 29 January 2018; Revised 27 March 2018; Accepted 12 April 2018; Published 21 June 2018

Academic Editor: Hyung-Sup Jung

Copyright © 2018 Aiping Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The logging while drilling (LWD) electromagnetic weak signal detection model of a Duffing oscillator based on a one-dimensional nonlinear oscillator is established. The influences of noise on Duffing oscillator dynamic behavior of periodic driving force are discussed and evaluate the oscillator weak signal detection mechanism based on a phase change trajectory diagram. The improved Duffing oscillator is designed and applied to detect the electromagnetic logging signal resistivity at the drill bit using the time-scale transformation method. The simulation results show that the nonlinear dynamic characteristics of the Duffing oscillator are very noticeable, the Duffing circuit is very sensitive to detect the tested signal, and it has a reasonable level of immunity to noise. The smaller the amplitude of the tested signal, the more sensitive the circuit is to the signal, the better the antinoise system performance, and the lower the signal-to-noise ratio (SNR).

#### 1. Introduction

During the logging while drilling (LWD) process, the drilling environments pose extreme challenges on the course of data detection, consequently changing the drilling signal properties and disrupting the surface signal detection, extraction, and interpretation. Accurate drilling information determines the correct directional drilling variables and geological formation from the wellbore to improve drilling efficiency [1, 2]. Drilling signal pulses experience unpredictable and complex adjustable signal impairments caused by friction between drill bit and stratum formation, drill string vibration noises, signal attenuation and drilling fluid circulations, high temperature, and high pressure of the downhole environment [3–6]. The impact leads to profound signal disruption and noise sources at the surface measurement system, resulting in surface signal detection complexity. On the other hand, the penetration of the drilling fluid causes the formation of mud cake, flushing zone, and transition zone along the radial direction of the borehole wall, and finally the undisturbed formation contains stratum information transmitted as signals to be detected by the surface receiver system [7, 8]. Greater well depths along the radial LWD equipment direction require depth drilling instruments which increase the signal transmission distances, consequently resulting in extremely weaker surface-detected signals. Therefore, the study of weak signal detection in the presence of strong noise is vital to improve the detection performance of logging tools.

Based on the chaotic nonlinear theory, this paper studies the weak signal detection method of a Duffing oscillator with azimuth electromagnetic wave logging resistivity. Using a one-dimensional nonlinear oscillator model, the Duffing oscillator detection model is established. The dynamic simulation analysis was carried out to study the influence of noise on the Duffing oscillator movement. The Duffing oscillator circuit is extended to improve its application into electromagnetic resistivity LWD signal detection based on the time-scale transformation technique. The paper discusses the use of circuit simulation to realize physical Duffing oscillator performance on the azimuth electromagnetic logging weak signal detections.

The rest of the paper proceeds as follows: Section 2 discusses the weak signal detection techniques, Section 3 describes the general and improved Duffing oscillator circuit and the influence of noise on Duffing oscillator movement, Section 4 discusses the simulation results of various Duffing oscillator phase state performances, and Section 5 discusses the paper summary while Section 6 draws the paper conclusion.

#### 2. Weak Signal Detection Techniques

Modern signal processing techniques for weak signal detection methods, principally, apply the conventional methods based on time- and frequency-domain techniques. Conventional time-domain weak signal detection methods mostly implement the correlation detection method [9] and sampling integral and time-domain averaging method [10, 11], such as those using the locking amplifier and sampling integrator. Frequency-domain methods are based on the signal power spectrum analysis [12] but are more suitable for weak periodic signal detections. Other weak signal detection methods deploy wavelet analysis [13, 14], high-order spectrum analysis [15, 16], Hilbert-Huang transform [17], and artificial neural network [18]. However, in the wavelet analysis, the selection of the wavelet base function and the scale range are very complex and lack the general method. High-order spectrum statistics analysis is an effective stochastic signal analysis tool that can effectively solve the objective non-Gaussian and nonlinear problems.

Initially, locally optimum detectors using the Neyman-Pearson approach performance criterion based on the additive noise model have been deployed to provide correct weak signal detection for the fixed value probability of false detection or false alarm [19]. The design used known signal sequence in which the observations represent either noise only or a signal with additive noise and specific sample size signal value. The additive noise models used approximation of actual circumstances and are not convenient to implement under certain consideration. Nonadditive noise models are necessary and cannot be ignored in multiplicative and signal-dependent noise communications, acoustics, and image processing applications [20]. The level of the contribution of higher-order statistics or of nonlinearity is not negligible. The nonadditive noise model to produce realistic and reasonable weak signal detection approximations has to be devoted to overcome the significant penalty which results from using the incorrect (additive noise) model. Most modem signal detectors based on statistical hypothesis testing assume the digital domain processing the useful and inevitable quantizer-detector model in modem signal detection problems.

Signal processing of weak signal detection using narrow-band high-order statistics attained a certain effect in the context of Gaussian noise but was only capable to inhibit white noise and Gaussian noise and took very large amount of calculations [15]. Neural network analysis uses an algorithmic mathematical model that simulates the neural structure of biological organisms to carry out distributed parallel information processing. By adjusting the interconnection relationship between large numbers of nodes, the neural network can be used to achieve information processing. Signal detection based on the neural network algorithm imposes setting the weight and critical values that need a lot of sample training. In case the noise type or magnitude changes, the neural network must be retrained; this limits the application of the neural network in the weak signal detection.

The traditional weak signal detection method estimates the noise signals to be the interfering signal whereby the original weak signal detection is achieved by noise suppression techniques. However, during noise suppressions, inevitably, useful signal is distorted, as this is a detrimental detection technique. In this situation, weak signal detection researchers began to carry out study on techniques to detect weak signal without destroying the useful signal in the low SNR conditions based on utilizing the noise rather than on suppressing it. In recent years, research and development of nonlinear science has provided novel weak signal detection techniques.

#### 3. Proposed Method

##### 3.1. Duffing Oscillator Model

The vibration law of the one-dimensional periodic driving force vibration system can be expressed mathematically as a differential equation shown by where “·” and “··” represent the first and second derivative of time , respectively; , , and represent the acceleration, velocity, and displacement of the particle, respectively; represents the damping force; represents the resilience; and represents the periodic driving force.

In the study of engineering kinetic technology, the calculation and analysis of nonlinear vibration characteristics describe the Duffing nonlinear oscillator with the cubic derivative term for the spring effect of mechanical complications. The proposed standardized Duffing equation becomes where represents the damping ratio coefficient, and are the linear and nonlinear stiffness coefficients, respectively, of the spring, and represents the amplitude of the periodic driving force.

On this basis, Moon and Holmes modify the Duffing equation to describe the support beam forced vibration in the inhomogeneous (nonuniform) magnetic field of the two permanent magnets, named as Duffing-Holmes described by

From (3), the magnitude of the damping ratio coefficient, , and the amplitude of the periodic driving force, , are variable parameters, while is defined as the nonlinear restoring force and named as the external periodic dynamic force. The Duffing equation has strong dynamic behavior, with noticeable nonlinear dynamic characteristics, creating one of the typical nonlinear dynamic system behavior applications.

In order to facilitate the analysis of the Duffing equation and achieve the purpose of reducing the order, (3) can be expressed as the differential equation:

##### 3.2. The Influence of Noise on Duffing Oscillator Movement

Due to the complex downhole conditions, the logging signal transmission process is disturbed by all kinds of noises which causes the surface-collected signal to contain many signal noises that weaken the amplitude and hence challenge the signal detection. Assuming that is the background noise in the detection process with zero expectation and the variance is , then, the solution of (3) in the critical state has a small change, , to the detection system output . Therefore, the system detection equation can be expressed as:

Subtracting (5) from (3), the new generated equation becomes

The variable change, , is very small compared to the system output detection . Therefore, by ignoring the high order of , the vector differential equation changes to

Each vector in the differential equation (7) can individually be expressed as

According to the solution existence and uniqueness theorem, the vector differential equation (7) has a unique solution that satisfies an initial condition which can be expressed as

Due to the performance analysis of the system at a steady state from (9), is the state matrix of the system. The first item is a transient solution, so the phase trajectory decays exponentially to 0 at time, , and the second item (statistical characteristic analysis) provides the expectation (mean) expressed as

The matrix variance can further be simplified and expressed as

The dynamic characteristics of the Duffing oscillator analysis show that as long as the mathematical expectation of the noise is zero, the noise has no decisive change. This concludes that the noise does not fundamentally change the phase of the system but only makes the phase trajectory of the system rough; the roughness is determined by the variance of the disturbance, but the overall system mean is zero. Additionally, since the noise distribution problem in the above analysis process is not principally limited, the Duffing oscillator has a good immune performance for any randomly distributed stochastic noise.

##### 3.3. Improved Duffing Oscillator

The traditional Duffing equation applies only to low frequency ranges while electromagnetic logging coil waves use high frequencies such as 2 MHz, 1 MHz, and 400 kHz. In order to apply the advanced weak signal detection technique based on electromagnetic LWD signal data acquisition systems, the improved dynamic equation that uses phase trajectory time-scale transformation to detect high-frequency signal can be expressed as

Assuming , computing becomes .

Equation (13) can further be simplified to

Substituting (13) and (14) into (12), the new simplified equation changes to

In order to facilitate the analysis of the equation, while achieving the purpose of reducing the order, (15) can be expressed by the differential state equations as follows:

##### 3.4. Response Signal Amplitude Detection

The method of detecting the amplitude of the signal using the Duffing oscillator uses system characteristics sensitive to the signal parameter to be tested and immune to noise signal. The basic detection idea is to set the frequency of the internal power (driving force) signal to be the same as the frequency of the signal to be measured and then adjust the amplitude of the internal driving force signal to make the system in chaotic critical state. The applied periodic signal to the Duffing oscillator system is used as the source signal and the internal driving force which must be in the same frequency and phase as the internal power signal. Increase the driving force magnitude until the system changes from the chaotic state into the large-scale periodic state and then gradually reduce the amplitude of the internal power until the system enters again into the chaotic critical state, and record the internal driving force amplitude at this time, . The amplitude of the signal to be measured and detected, , is computed as the amplitude difference between the magnitudes of the internal driving force when the system is in the critical state and when it enters the chaotic state from the large periodic state, that is, .

By automatically adjusting the amplitude of the internal driving force and identifying the phase, the signal amplitude can be detected and determined by the block diagram shown in Figure 1. The modified Duffing-Holmes equation is used to simulate the model; the vibrator (oscillator) movement state is related to the magnitude of the damping coefficient ratio, , and the amplitude of the excitation force, , of the periodic driving force. Any changes or disturbances into these parameters can obviously change the output phase of the system. The damping coefficient and the power of the disturbance can be observed during signal detection. The signal is detected by consistent disturbance of the measured signal damping coefficient ratio and the amplitude of the power signal amplitude in a chaotic critical state.