**Wavelet-Based Analysis of MCSA for Fault Detection in Electrical Machine**

Mohammad Rezazadeh Mehrjou, Norman Mariun, Mahdi Karami, Samsul Bahari Mohd. Noor, Sahar Zolfaghari, Norhisam Misron, Mohd Zainal Abidin Ab. Kadir, Mohd. Amran Mohd. Radzi and Mohammad Hamiruce Marhaban

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/61532

#### **Abstract**

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78 Wavelet Transform and Some of Its Real-World Applications

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Early detection of irregularity in electrical machines is important because of their diversity of use in different fields. A proper fault detection scheme helps to stop the propagation of failure or limits its escalation to severe degrees, and thus it prevents unscheduled down‐ times that cause loss of production and financial income. Among different modes of fail‐ ures that may occur in the electrical machines, the rotor-related faults are around 20%. Successful detection of any failure in electrical machines is achieved by using a suitable condition monitoring followed by accurate signal processing techniques to extract the fault features. This article aims to present the extraction of features appearing in current signals using wavelet analysis when there is a rotor fault of eccentricity and broken rotor bar. In this respect, a brief explanation on rotor failures and different methods of condition moni‐ toring with the purpose of rotor fault detection is provided. Then, motor current signature analysis, the fault-related features appeared in the current spectrum and wavelet trans‐ form analyses of the signal to extract these features are explained. Finally, two case studies involving the wavelet analysis of the current signal for the detection of rotor eccentricity and broken rotor bar are presented.

**Keywords:** Wavelet transform, Line start permanent magnet motor, Induction motor, Ec‐ centricity, Broken rotor bar

#### **1. Introduction**

Electrical machines are widely used for many industrial processes and play a non-substitutable role in a variety of industries [1–2]. In spite of their reliability and robustness, electrical

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

machines are still prone to failures due to the exposure to a wide diversity of strict conditions and environments, incorrect operations, or even manufacturing defects [3]. These faults, gradual deterioration, and failures can lead to motor interruption, if left undetected, and their resulting unplanned downtime is very expensive. Early detection of irregularity in electrical machines with proper fault diagnosis schemes will help prevent high-cost failures, thereby decreasing the cost of maintenance and preventing unscheduled downtimes. However, stopping the propagation of the fault limits its escalation to severe degrees, which results in loss of production and financial income.

The fault identification schemes are basically based on data collection of electrical machines followed by signal processing. The condition of an electrical machine is examined from data that are acquired through sensors and supportive equipment methods. By using a suitable signal processing technique, each fault can be then detected via a specific feature present in the measured signal of a faulty motor when compared to a fault-free one. Hitherto, a number of data acquisition techniques, which display a certain parameter of the electrical machine, have been established. Essentially, the efficiency of a data acquisition method is characterized by its accuracy, cost, and importantly its capability to quantify the fault. On the contrary, condition monitoring based on data acquisition techniques requires the user to have adequate knowledge and proficiency to differentiate a normal operating condition from a potential failure state. The key step, but a difficult task, in the fault detection of electrical machines is to extract the fault-related features from the acquired signal and identify the condition of motor. Fault-related features are parameters derived from the acquired data that specify the existence of failure in device. The current signal of electrical machine is non-linear and non-stationary with strong noise interference; hence, the energy of early signal is too low to extract faultrelated features in time domain [4]. Advanced signal processing methods based on analysis of time–frequency domain have been proposed as effective approaches for fault detection in electrical machines [5].

The focus of this chapter is on the extraction of features present in the current spectrum of electrical machine when one of two important rotor faults, eccentricity and broken rotor bar, exists. To extract features from the current spectrum in the presence of these faults, an advanced signal processing method, wavelet packet analysis, is used. In this regard, the fundamentals related to the detection of these two faults using wavelet packet analysis of current signal are explained in the following sections.

#### **2. Rotor faults**

From the investigations on different failure modes in electrical machines, the rotor-related faults are around 20% of failures may happen in the motor [6]. The rotor is exposed to different types of stresses that seriously affect its normal condition and subsequently create faults in it. Bonnett and Soukup explained the stresses that motors are subjected to and their unfavourable causes [7]. Failures in rotor are classified into eccentricity of rotor, crack and/or breakage of rotor cage bars, and crack and/or breakage of end rings and rotor bow [8]. These irregularities bring specific secondary failures that cause serious faults in electrical machines. Moreover, these types of faults may not show any symptoms during early stage until propagating to the next step and leading to the sudden collapse [9–11]. In recent years, rotor faults have been increasingly studied for developing advanced techniques that permit online early detection and diagnosis of motor faults to avoid any negative consequences of unexpected shutdowns, but this area still needs more research because of the complexity of the motor during the runtime. In this section, a brief description of different rotor faults is provided.

#### **2.1. Rotor eccentricity**

machines are still prone to failures due to the exposure to a wide diversity of strict conditions and environments, incorrect operations, or even manufacturing defects [3]. These faults, gradual deterioration, and failures can lead to motor interruption, if left undetected, and their resulting unplanned downtime is very expensive. Early detection of irregularity in electrical machines with proper fault diagnosis schemes will help prevent high-cost failures, thereby decreasing the cost of maintenance and preventing unscheduled downtimes. However, stopping the propagation of the fault limits its escalation to severe degrees, which results in

The fault identification schemes are basically based on data collection of electrical machines followed by signal processing. The condition of an electrical machine is examined from data that are acquired through sensors and supportive equipment methods. By using a suitable signal processing technique, each fault can be then detected via a specific feature present in the measured signal of a faulty motor when compared to a fault-free one. Hitherto, a number of data acquisition techniques, which display a certain parameter of the electrical machine, have been established. Essentially, the efficiency of a data acquisition method is characterized by its accuracy, cost, and importantly its capability to quantify the fault. On the contrary, condition monitoring based on data acquisition techniques requires the user to have adequate knowledge and proficiency to differentiate a normal operating condition from a potential failure state. The key step, but a difficult task, in the fault detection of electrical machines is to extract the fault-related features from the acquired signal and identify the condition of motor. Fault-related features are parameters derived from the acquired data that specify the existence of failure in device. The current signal of electrical machine is non-linear and non-stationary with strong noise interference; hence, the energy of early signal is too low to extract faultrelated features in time domain [4]. Advanced signal processing methods based on analysis of time–frequency domain have been proposed as effective approaches for fault detection in

The focus of this chapter is on the extraction of features present in the current spectrum of electrical machine when one of two important rotor faults, eccentricity and broken rotor bar, exists. To extract features from the current spectrum in the presence of these faults, an advanced signal processing method, wavelet packet analysis, is used. In this regard, the fundamentals related to the detection of these two faults using wavelet packet analysis of

From the investigations on different failure modes in electrical machines, the rotor-related faults are around 20% of failures may happen in the motor [6]. The rotor is exposed to different types of stresses that seriously affect its normal condition and subsequently create faults in it. Bonnett and Soukup explained the stresses that motors are subjected to and their unfavourable causes [7]. Failures in rotor are classified into eccentricity of rotor, crack and/or breakage of rotor cage bars, and crack and/or breakage of end rings and rotor bow [8]. These irregularities

loss of production and financial income.

80 Wavelet Transform and Some of Its Real-World Applications

electrical machines [5].

**2. Rotor faults**

current signal are explained in the following sections.

In a fault-free machine, the rotor is centre aligned in the stator bore that results in uniform air gap between the stator and rotor. In fault-free electrical machines, the rotation centre of the rotor is the same as the geometric centre of the stator bore. As a result, the rotor symmetrical axis (*C*<sup>r</sup> ), stator symmetrical axis (*C*s), and rotor rotational axis (*C*g) coincide with each other, and thus the magnetic forces are balanced in opposite directions. Rotor eccentricity, displace‐ ment of the rotor from its centred position in the stator bore, generates an asymmetric air gap between the stator and rotor [12]. The rotor eccentricity also produces unbalanced magnetic pull (UMP), which is a radial magnetic force on the rotor shaft. The UMP also pulls away the rotor from the stator bore centre, thus causing excessive stress on the electrical machine [13, 14]. Eccentricity commonly presents in rotating electrical machines, and the maximum permissible level of eccentricity is defined, which is 5 or 10% of the air-gap length [15]. If eccentricity exceeds the permissible level, it will increasingly damage the winding, stator core and rotor core in the motor due to rubbing of the stator with the rotor [12, 14]. Three different types of eccentricity occur in an electrical machine: static eccentricity, dynamic eccentricity, and mix eccentricity. As an example, static eccentricity is explained next.

Static eccentricity in electrical machines occurs when the rotor symmetrical axis is concentric with the rotor rotational axis; however, they are dislocated with respect to the stator symmet‐ rical axis; hence, the position of minimum radial air-gap length is fixed. In this state, the mutual inductances across the stator and rotor as well as the self- and mutual inductances among the rotor phases are related to the angular position of the rotor [16]. The implication of static eccentricity fault in motor is depicted in Figure 1.

Static eccentricity can be due to numerous motives such as elliptical stator core, wrong placement of the rotor or stator at the setup or subsequent of maintenance, incorrect bearing positioning, bearing deterioration, shaft deflection, housing imperfection, end-shield mis‐ alignment, excessive tolerance, and rotor weight or pressure of interlocking ribbon [17–19]. Static eccentricity leads to second failures which cause drastic harm to the rotor, stator core and windings. The radial forces in the static eccentricity condition produce a steady UMP in the radial route across the motor because the reluctance of the magnetic flux path decreases with the transmission of flux on the side of tiny air gap [20]. Albeit, the winding current induces more magnetic flux that causes a stronger pull and leads to the expansion of the air gap on the opposite side where the reluctance increases, thereby decreasing the flux and magnetic side pull. Therefore, the UMP compels the rotor to move toward the area of the narrowest air-gap length. During abrasion, the stator core subsequently generates abnormal vibration and severely damages the rotor, windings and the stator [7]. Consequently, the static eccentricity causes acoustic noise, premature failure in the bearing, rotor deflection and bent rotor shaft.

**Figure 1.** Cross section of motor under static eccentricity fault

The degree of static eccentricity is calculated by the equation based on Figure 2 [16]:

$$D\_{\rm SE} = \frac{\left| \overline{\mathbf{C}\_{\rm s} \mathbf{C}\_{\rm g}} \right|}{\mathcal{g}} \tag{1}$$

where CsCg → is the vector of static transfer which is invariant for rotor angular positions and *<sup>g</sup>* is the uniform air-gap length.

**Figure 2.** Location of stator and rotor under static eccentricity condition

#### **2.2. Rotor bar breakage**

severely damages the rotor, windings and the stator [7]. Consequently, the static eccentricity causes acoustic noise, premature failure in the bearing, rotor deflection and bent rotor shaft.

The degree of static eccentricity is calculated by the equation based on Figure 2 [16]:

uuuuur s g

CC

*g*

→ is the vector of static transfer which is invariant for rotor angular positions and *<sup>g</sup>*

=

SE

*D*

**Figure 1.** Cross section of motor under static eccentricity fault

82 Wavelet Transform and Some of Its Real-World Applications

**Figure 2.** Location of stator and rotor under static eccentricity condition

where CsCg

is the uniform air-gap length.

The breakage of rotor bars is one of the important failures in the rotor cage of electrical machines. During the operation of electrical machines, rotor bars may be broken partially or completely. The main reasons for bar breakage include electrical, mechanical, and environ‐ mental stresses during the operation of electrical machines and/or improper design of rotor geometry. Once a bar breaks, the stress increases and deteriorates the condition of the neighbouring bars progressively. Such a destructive process can be prevented, if any crack in the bar is detected early [21]. Typical causes of rotor bar breakage are referred as follows [22]: high thermal and mechanical stresses, direct online starting duty cycles for which the rotor cage was not well designed to endure against the stresses, imperfections in design and fabrication process of the rotor cage bars. Any failure in rotor bars itself causes unbalanced currents and torque pulsation and, therefore, decreases the average torque [23].

The rotor bars are short-circuited on both sides of the rotor by end rings. Depending on the type of squirrel cage in the motor, the source of failures in the end ring differs, in die-cast aluminium rotors caused by porosity of casting and in fabricated rotor cages caused by poor end-ring joints during manufacturing. Once the preliminary failure occurs, localized heat may extend to the rotor cage excessively. Therefore, the fault propagation is continued by multiple start-ups similar to load variations, which create high centrifugal forces. Accordingly, end-ring faults cause a drastic increase in the current and speed fluctuation [21].

#### **2.3. Rotor bow**

(1)

Any irregular thermal variation (heating or cooling) and unfavourable thermal distribution of the rotor during operation of electrical machines may bow the rotor [24]. The bow created in the rotor prevents sufficient alignment in the motor and generally produces a preload on the bearings. Bend locations in the rotor cause major failures in other parts of the motor [25]. The bow in the rotor is classified as local and extended [24]. When an asymmetrical heating is confined to a part of the rotor, a local bow is generated. For example, rotor-to-stator rubbing can generate a local asymmetric thermal distribution, which causes the local bow. When an asymmetrical heating extends along the rotor, an extended bow is generated. Long-lasting gravity effects on off-line machines generate rotor bow classified as an extended bow, when unsuitable rotor straightening turning system is not used [24]. Since the rotor is limited by two bearings, extended bow commonly causes a shaft bow [24].

#### **3. Condition monitoring techniques for rotor fault detection**

Condition monitoring programme which can predict a failure in electrical machines has received considerable attention for many years [2, 8]. Successful detection of any failure in electrical machines is achieved by using suitable condition monitoring. When a failure occurs, some machine parameters are exposed to changes that depend upon the fault degree. Any irregularity in the rotor of electrical machines presents with variation distributed in the rotor currents. The feedback of these currents to the air-gap field produces specific signatures of fault in the spectrum of speed, torque, current, and power. Reliable condition monitoring techniques depend on the best understanding of the mechanical and electrical characteristics of the electrical machines in both fault-free and faulty situations. Researchers have used different condition monitoring techniques that can be categorized as follows [8]:


#### **3.1. Motor current signature analysis**

The drowned current signal by an electrical machine contains a single component. Any magnetic or mechanical asymmetries in the machine generate other frequency components in the stator current spectrum. These frequency components are diverse according to each specific fault in the machine.

Motor current signature analysis (MCSA) analyses the stator current signal to identify the presence of any failure in electrical machines. This analysis method has been introduced as an effective way for monitoring electrical machines for many years [8]. From all these methods suggested in the literature, MCSA is a forerunner because of its advantages [10, 13, 26–28]:


When a failure is generated in the electrical machine, depending on the severity of this fault, some of the machine parameters change. For instance, the current spectrum of an ideal electrical machine contains a single component corresponding to the supply frequency. Any asymmetry in electrical machine causes other components to appear in a spectrum of stator current. When a rotor bar breaks, current does not flow through it, and hence no magnetic flux is created around the breakage bar. Therefore, there is no non-zero backward rotating field that rotates at the slip frequency speed with respect to the rotor. This asymmetry in the magnetic field of rotor induces harmonics in stator windings, which are superimposed on it. These superimposed harmonics appear at frequency spectrums as described in

$$f\_{\text{BRB}} = \left[1 \pm 2 \text{KS}\right] f\_{\text{S}} \tag{2}$$

where *f* BRB is the harmonic component due to broken rotor bar, S is the slip, *f*S is the funda‐ mental frequency, and *k* = 1, 2,... [29].

Any asymmetry caused by static eccentricity produced other components that appear in the spectrum of stator current. The characteristic frequency component associated with static eccentricity is located according to Eq. (3) [16]:

$$f\_{\text{static}} = \left[1 \pm \frac{m}{p}\right] f\tag{3}$$

where *f* static is the harmonic component due to static eccentricity in line start permanent magnet synchronous motor (LSPMSM), *m* is an odd integer value, *p* is the number of pole pair, and *f* is the line frequency.

#### **4. Wavelet**

techniques depend on the best understanding of the mechanical and electrical characteristics of the electrical machines in both fault-free and faulty situations. Researchers have used

The drowned current signal by an electrical machine contains a single component. Any magnetic or mechanical asymmetries in the machine generate other frequency components in the stator current spectrum. These frequency components are diverse according to each specific

Motor current signature analysis (MCSA) analyses the stator current signal to identify the presence of any failure in electrical machines. This analysis method has been introduced as an effective way for monitoring electrical machines for many years [8]. From all these methods suggested in the literature, MCSA is a forerunner because of its advantages [10, 13, 26–28]:

**•** Different fault detection capability (such as broken rotor bars, air-gap eccentricity, stator

different condition monitoring techniques that can be categorized as follows [8]:

**•** Acoustic emission

**•** Electromagnetic field monitoring

84 Wavelet Transform and Some of Its Real-World Applications

**3.1. Motor current signature analysis**

**•** Online monitoring characteristics

**•** Inexpensive equipment and easy measurement

**•** Remote monitoring ability

**•** Early-stage fault detection

**•** Non-invasive feature

faults, etc.)

**•** Highly sensitive

**•** Selective

**•** Instantaneous angular speed

**•** Motor circuit analysis

**•** Air-gap torque

**•** Induced voltage

**•** Current

**•** Power

**•** Surge testing

fault in the machine.

**•** Vibration

**•** Voltage

Frequency domain analysis is not reliable for fault detection because some outside parameters can affect the location and amplitude of fault-related feature. This parameter can be classified as follows: first, the fault frequency components depend on the slip of the motor; second, the fault feature amplitude is load dependent; third, the frequencies of the fault components are affected by voltage fluctuations; and fourth, long sampling interval is needed for a highresolution frequency. Therefore, in general, frequency domain analyses are suitable for the steady-state situation. The problem involved in the analysis of non-stationary signals can be shunned by time–frequency analysis of the signal, which illustrates the signal in threedimensional axis as time, frequency, and amplitude. The most popular time–frequency representations include Wigner–Ville distribution, short-time Fourier transform, and wavelet transform.

Wavelet transform expresses a signal in oscillatory function series at different frequencies and time. Wavelet transform divides the original signal into time-scale space, where the dimension of windows at time and scale (frequency) is not rigid [30]. Therefore, in fault diagnostics domain, wavelet transform has been used to extract the dominant features from original signals [31]. Various types of wavelet transforms have been widely used in the condition monitoring of electrical machine. Among all these techniques, discrete wavelet transform and wavelet packet transform (WPT) are the most common ones, explained in the following.

#### **4.1. The principle of discrete wavelet decomposition**

Discrete wavelet transform is based on signal analyses using a minor set of scales and specific number of translations at each scale. Mallat (1989) introduced a practical version of discrete wavelet transform called wavelet multi-resolution analysis [32]. This algorithm is based on the fact that one signal is disintegrated into a series of minor waves belonging to a wavelet family.

A discrete signal *f*[*t*] could be decomposed as

$$f\left[\underline{t}\right] = \sum\_{k} A\_{m0,n} \phi\_{m0,n} \left[\underline{t}\right] + \sum\_{m=m0}^{m-1} \sum\_{n} D\_{m,n} \nu\_{m,n} \left[\underline{t}\right] \tag{4}$$

where *ϕ* is the scaling function (father wavelet) and *ψ* is the wavelet function (mother wavelet), *A* is the approximate coefficient and *D* is the detail coefficient.

The multi-resolution analysis commonly uses discrete dyadic wavelet, in which positions and scales are based on powers of two. In this approach, the scaling function is depicted by the following equation:

$$\left[\phi\_{m0,n}\right]\left[t\right] = 2^{m0/2}\phi\left(2^{m0}t - n\right) \tag{5}$$

that is, *ϕm*0,*n* is the scaling function at a scale of 2*<sup>m</sup>*<sup>0</sup> shifted by *n*. Wavelet function is also defined as

$$\mathbb{E}\left[\boldsymbol{\nu}\_{m,n}\Big|\boldsymbol{t}\right] = \mathbb{Z}^{m/2}\mathbb{1}\left(\mathbb{Z}^m\boldsymbol{t} - n\right) \tag{6}$$

that is, *ψm*,*<sup>n</sup>* is the mother wavelet at a scale of 2*<sup>m</sup>* shifted by *n*.

Generally, approximate coefficients *Am*0,*<sup>n</sup>* are obtained through the inner product of the original signal and the scaling function

Wavelet-Based Analysis of MCSA for Fault Detection in Electrical Machine http://dx.doi.org/10.5772/61532 87

$$A\_{m0,u} = \bigcap\_{-\infty}^{\infty} f\left(t\right) \phi\_{m0,u}\left(t\right) dt \tag{7}$$

The approximate coefficients decomposed from a discretized signal can be expressed as

Wavelet transform expresses a signal in oscillatory function series at different frequencies and time. Wavelet transform divides the original signal into time-scale space, where the dimension of windows at time and scale (frequency) is not rigid [30]. Therefore, in fault diagnostics domain, wavelet transform has been used to extract the dominant features from original signals [31]. Various types of wavelet transforms have been widely used in the condition monitoring of electrical machine. Among all these techniques, discrete wavelet transform and wavelet packet transform (WPT) are the most common ones, explained in the following.

Discrete wavelet transform is based on signal analyses using a minor set of scales and specific number of translations at each scale. Mallat (1989) introduced a practical version of discrete wavelet transform called wavelet multi-resolution analysis [32]. This algorithm is based on the fact that one signal is disintegrated into a series of minor waves belonging to a wavelet family.


*m m nmn mn mn*

1 0, 0. , , 0

= é ù = + é ù é ù ë û å åå ë û ë û

where *ϕ* is the scaling function (father wavelet) and *ψ* is the wavelet function (mother wavelet),

The multi-resolution analysis commonly uses discrete dyadic wavelet, in which positions and scales are based on powers of two. In this approach, the scaling function is depicted by the

> f

 y

ë û = - ( ) / 2 , 2 2 *m m*

Generally, approximate coefficients *Am*0,*<sup>n</sup>* are obtained through the inner product of the

é ù

ë û <sup>=</sup> ( - ) 0/2 0 0, 2 2 *m m*

é ù

 y

*ft A t D t* (4)

*m n t t n* (5)

*m n t tn* (6)

shifted by *n*. Wavelet function is also defined

f

*A* is the approximate coefficient and *D* is the detail coefficient.

f

y

that is, *ψm*,*<sup>n</sup>* is the mother wavelet at a scale of 2*<sup>m</sup>* shifted by *n*.

that is, *ϕm*0,*n* is the scaling function at a scale of 2*<sup>m</sup>*<sup>0</sup>

original signal and the scaling function

*k mm n*

**4.1. The principle of discrete wavelet decomposition**

A discrete signal *f*[*t*] could be decomposed as

86 Wavelet Transform and Some of Its Real-World Applications

following equation:

as

$$A\_{\left(m+1\right),n} = \sum\_{n=0}^{N} A\_{m,n} \left[ \phi\_{m,n}\left(t\right) \phi\_{m+1,n}\left(t\right) dt = \sum A\_{m,n} \cdot g\left[m\right] \tag{8}$$

In the dyadic approach, the approximation coefficients *Am*0,*n* are at a scale of 2*<sup>m</sup>*<sup>0</sup> . The filter, *g[n]*, is a low-pass filter. Similarly, the detail coefficients *Dm*,*<sup>n</sup>* can be generally obtained through the inner product of the signal and the complex conjugate of the wavelet function:

$$D\_{m,n} = \bigcap\_{-\infty}^{\infty} f(t). \varphi\_{m,n}^\* \left( t \right) dt \tag{9}$$

The detail coefficients decomposed from a discretized signal can be expressed as

$$D\_{\left(m+1\right),n} = \sum\_{n=0}^{N} A\_{m,n} \left[ \phi\_{m,n}\left(t\right) \mu\_{m\ast 1..n}\left(t\right) dt = \sum A\_{m,n} h\left[n\right] \right] \tag{10}$$

In the dyadic approach, *Dm*,*n* are the detail coefficients at a scale of 2*<sup>m</sup>*<sup>0</sup> . The filter, *h[n]* is a high-pass filter.

The multi-resolution analysis utilizes discrete dyadic wavelet and extract approximations of the original signal at different levels of resolution. An approximation is a low-resolution representation of the original signal. The approximation at a resolution 2<sup>−</sup>*<sup>m</sup>* can be split into an approximation at a coarser resolution 2<sup>−</sup>*m*−<sup>1</sup> and the detail. The detail represents the highfrequency contents of the signal. The approximations and details can be determined using lowand high-pass filters. In the multi-resolution analysis, the approximations are split successively, while the details are never analysed further. The decomposition process can be iterated, with successive approximations being decomposed in turn; hence one signal is broken down into many lower-resolution components. This process is called the wavelet decompo‐ sition tree as shown in Figure 3. It illustrates the dyadic wavelet decomposition algorithm regarding the coefficients of the transform at different levels according to the description by Polikar et al. (1998) [33].

**Figure 3.** Dyadic wavelet decomposition algorithm [34].

#### **4.2. The principle of wavelet packet decomposition**

The wavelet packet transform is a direct expansion of discrete wavelet transform, where the details as well as approximation are split up. Therefore, this tree algorithm is a full binary tree that offers rich possibilities for signal processing and better signal representation in compari‐ son to a discrete one.

A wavelet packet function has three naturally interpreted indices in time–frequency functions:

$$\boldsymbol{\Psi}\_{j,k}^{\boldsymbol{i}} = 2^{\frac{\boldsymbol{i}}{2}} \boldsymbol{\Psi}^{\boldsymbol{i}} \left( 2^{\boldsymbol{i}} \boldsymbol{t} - \boldsymbol{k} \right), \; \boldsymbol{i} = \mathbf{1}, \; \mathbf{2}, \; \mathbf{3}, \dots \tag{11}$$

where integers *j*, *k,* and *i* are called the scale, translation, and simulation parameters, respec‐ tively. Scaled filter *h*(*n*) and the wavelet filter *g*(*n*) are quadrature mirror filters associated with the scaling function *Φ*(*t*) and the wavelet function *ψ*(*t*) [32]. The conjugate mirror filters *h* and *g* with finite impulse responses (FIRs) of size *k* can define the fast binary wavelet packet decomposition (WPD) algorithm of the signal *f*(*t*):

$$\begin{cases} d\_0^0 \left( t \right) = f\left( t \right) \\ d\_{j+1}^{2n} \left( t \right) = \sum\_{k} h\left( k - 2t \right) d\_j^i \; i = 0, 1, \dots, 2^j - 1 \\ \qquad d\_{j+1}^{2n+1} \left( t \right) = \sum\_{k} \mathbf{g}\left( k - 2t \right) d\_j^i \end{cases} \tag{12}$$

The wavelet packet component signals *f <sup>j</sup> i* (*t*) are produced by a combination of wavelet packet function *ψj*,*<sup>k</sup> <sup>n</sup>* (*t*) as follows:

$$\mathbf{f}\_{/}^{i}\left(\mathbf{t}\right) = \sum\_{l=1}^{2/p} \mathbf{C}\_{j,k}^{l}\left(\mathbf{t}\right) \boldsymbol{\nu}\_{j,k}^{i}\left(\mathbf{t}\right) \tag{13}$$

where the wavelet packet coefficients *C <sup>j</sup>*,*<sup>k</sup> <sup>i</sup>* (*t*) are calculated by

$$C\_{j,k}^{l}\left(t\right) = \bigcap\_{\cdots \neq \ }^{\circ}f\left(t\right)\nu\_{j,k}^{l}\left(t\right)dt\tag{14}$$

Provided the wavelet packet functions are orthogonal

**Figure 3.** Dyadic wavelet decomposition algorithm [34].

88 Wavelet Transform and Some of Its Real-World Applications

son to a discrete one.

**4.2. The principle of wavelet packet decomposition**

y

decomposition (WPD) algorithm of the signal *f*(*t*):

ì

ï

ï

î

+

2 1

 y = -= ( ) <sup>2</sup> , 2 2 , 1, 2, 3,...

( ) ( )

*d t hk tdi*

*j j k*

<sup>ï</sup> <sup>=</sup> <sup>ï</sup>

å

+ +

2 1 1

<sup>ï</sup> = - <sup>ï</sup>

*j i i j*

The wavelet packet transform is a direct expansion of discrete wavelet transform, where the details as well as approximation are split up. Therefore, this tree algorithm is a full binary tree that offers rich possibilities for signal processing and better signal representation in compari‐

A wavelet packet function has three naturally interpreted indices in time–frequency functions:

where integers *j*, *k,* and *i* are called the scale, translation, and simulation parameters, respec‐ tively. Scaled filter *h*(*n*) and the wavelet filter *g*(*n*) are quadrature mirror filters associated with the scaling function *Φ*(*t*) and the wavelet function *ψ*(*t*) [32]. The conjugate mirror filters *h* and *g* with finite impulse responses (FIRs) of size *k* can define the fast binary wavelet packet

( ) ( )

2 0, 1,.....,2 1

2

í =- = -

*n i j*

*d t ft*

0 0

( ) ( )

*n i j j k*

å

*d t gk td*

*j k tk i* (11)

(12)

$$
\mu \nu\_{j,k}^{\
u} \left( t \right) \nu\_{j,k}^{\
u} \left( t \right) = 0 \text{ if } m \neq n \tag{15}
$$

As data sets of wavelet packet coefficients increase in size, the energy principle is applied to current signals after WPT for fault location estimation [35].

#### **4.3. A review of wavelet decomposition for fault detection**

Different types of wavelet transform techniques have been widely used in algorithms designed for fault detection in electrical machines. Table 1 presents the common types of these techniques.




**Ref Year**

[36] 2001

[40] 2004

[41] 2005 MCSA

**Diagnostic Monitoring Techniques (MTs)**

90 Wavelet Transform and Some of Its Real-World Applications

Motor current signature analysis (MCSA)

[37] <sup>2002</sup> MCSA WT,park

[38] <sup>2002</sup> MCSA, voltage, speed

[39] 2003 MCSA WPD

Current (start-up) **Signal Processing**

Discrete wavelet transform (DWT) -

transform(PT) -

DWT -

fast fourier transform (FFT),


WPD

Wavelet packet decomposition (WPD)

**Classifier and Decisionmaking Tool**

**Purpose Achievement and Limitation**

achievable

current signal

faults

mode

To develop current monitoring procedure for BRB detection

To compare modelbased and signal-based approaches based on Park transform for BRB

To develop a modelbased diagnosis system for detection of various faults including BRB

To improve MCSA monitoring procedure for BRB and air-gap eccentricity detection

To improve the start-up current monitoring procedure for BRB detection using a filter that actively tracks the changing amplitude, phase and frequency to extract the fundamental from the transient

To improve MCSA monitoring procedure for the detection of various faults including

BRB

detection

Artificial neural network

Artificial neural network

fully loaded conditions are neither practical nor

A new approach in detection of BRB having only stator

The spectral decomposition obtained by the wavelet transform may be used to isolate different kinds of faults

The proposed system was shown effective in detecting early stages of different IM

This method does not require parameters such as speed or number of rotor bars. It is not load dependent and can be applied to IMs that operate continuously in the transient

The features of BRB and static eccentricity yield similar results in the wavelet analysis, but are different in Fourier analysis. Therefore the use of both types of analysis together can distinguish the faults

It provides feature representations of multiple frequency resolutions for

faulty modes



**Ref Year**

[48] 2007

[52] 2008

Current (start-up)

Induced

voltage FFT, WT -

[49] 2007 MCSA WPD -

[50] 2007 MCSA WPD -

[51] 2008 MCSA STFT, WT -

DWT Principle

component

**Diagnostic Monitoring Techniques (MTs)**

92 Wavelet Transform and Some of Its Real-World Applications

**Signal Processing**

**Classifier and Decisionmaking Tool**

> or number of levels of the decomposition) over

the diagnosis

To investigate the limitations and harmonics of the induced voltage after supply disconnection harmonics for BRB

detection

To detect incipient bearing fault via stator current analysis

To detect real-time fault for various disturbances in three-phase IM

To improve MCSA monitoring procedure for BRB and stator shorted turns detection

To develop transient current monitoring

**Purpose Achievement and Limitation**

fault diagnosis

voltage

WPD method

mother wavelet

Cover better analysis under various conditions and more tolerant frequency bands with

Selecting the optimal levels of decomposition and optimum

Wavelet decomposition is superior to STFT. Power spectral density for wavelet details was introduced as a merit factor for fault diagnosis. The proposed method can diagnose shorted turns and BRB in non-constant load– torque IM applications

Feature reduction and extraction using component

allows a correct diagnosis of a fault-free machine in some particular cases where Fourier analysis leads to an incorrect

Fourier transform did not provide information about fault severity and load variations. A method based on wavelet analysis of induced voltage spectrum was developed for BRB detection Limitation: Tests need to be carried out for fault-free motor to develop a baseline response. It is sensitive to changes in load, system inertia, rotor temperature and supply


Wavelet-Based Analysis of MCSA for Fault Detection in Electrical Machine http://dx.doi.org/10.5772/61532 95


**Table 1.** Summary of published paper with the aim of using wavelet transform for broken rotor bar and eccentricity fault detection

#### **5. Case study 1**

**Ref Year**

**Diagnostic Monitoring Techniques (MTs)**

94 Wavelet Transform and Some of Its Real-World Applications

[56] 2009 Vibration WPD

[57] 2009 Vibration WPD

[58] 2009 MCSA DWT -

[59] 2010 MCSA WT,PSD -

[60] 2011 MCSA FFT,WT -

[4] 2012 Vibration WPD,EMD

[61] 2013 MCSA Stationary WPD

[62] 2013 Vibration WPD,FFT

**Signal Processing**

**Classifier and Decisionmaking Tool**

Artificial Neural Network

Hybrid support machine

Artificial neural network

Multiclass support vector machines

Artificial neural network

To optimize gear failure identification using GAs

To compare a different wavelet family for BRB fault detection

To develop BRB detection methods based on MCSA

To propose a new method for early fault

To integrate the fine resolution advantage of WPD with the selfadaptive filtering characteristics of empirical mode decomposition (EMD) to early fault diagnosis

BRB feature extraction by SWPT under lowersampling rate

To classify fault and predict remaining useful

life

detection

and ANNs

To propose an intelligent method to diagnose rotating machinery failures

**Purpose Achievement and Limitation**

instantaneous magnitude of the stator-current signal

The technique determines the

wavelet', 'decomposition level' and 'number of neurons in

An accurate and quick faulttype estimation method by applying hybrid SVM to the energy criterion after WPA

The technique determines the

The method has the ability to detect BRB for both constant torque and for variable load

The approach has been proved to be effective to detect failures in its very early stages

Ability to extract weak signals and early fault detection of rotating machinery

Lower computation and cost without any effect on the performance of SWPT to

detect BRB

To deal with complex problems and non-linear

best mother wavelet

torque

best values 'mother

hidden layer'

The detection of static eccentricity in three-phase LSPMSM using motor current signature analysis is studied. A detailed description of experimental test rig used in this study and the method used for signal measurement and analysis is provided in the following.

#### **5.1. Experimental set-up**

The experimental test rig is shown in Figure 4. The tested motor for both fault-free and faulty (with static eccentricity) cases is a three-phase LSPMSM with the specification as mentioned in Table 2. The motor is directly fed by the grid power supply, while the stator windings are Y connected, and the current nominal value is 1.28 A. The LSPMSM is coupled to torque/speed sensor in order to measure the torque value in different operation conditions. On the other side, a mechanical load is provided by a DC-excited magnetic powder brake (MPB) coupled to torque/speed sensor. The specific load torque level could be furnished to the motor shaft by controlling the input dc voltage of MPB. This system is used to sample the stator current noninvasively when the motor is operated in the steady-state condition. Notably, only one phasecurrent signal is required to be recorded for the detection process in this study. The recorded signals are analysed by a computer-based signal processing program.

#### **5.2. Method**

The method proposed in this study for the creation of eccentricity fault in LSPMSM, by changing the original bearings of motor with a new set of bearing with larger inner diameter and smaller outer diameter, results in the creation of free space between the shaft and bearings and also between the bearings and the housing of end shields. Static eccentricity is created by fixing concentric inner rings between the new bearings and shaft on both ends of LSPMSM and non-concentric outer rings between the new bearings and housings of both end shields. The aforementioned strategy is used to create 33% and 50% static eccentricity in the motor discussed in the case study.

The current spectrum is stored with the sampling frequency ( *f* <sup>S</sup>) of 5 kHz over a total sampling period of 6.5 s, which allows the analysis of the signals with a minimum frequency of 0.15 Hz. Daubechies-24 (db24) is used as the mother wavelet in discrete wavelet transform (DWT) analyses. Since the four-pole, three-phase LSPMSM is considered, the characteristic frequency component associated with static eccentricity is located at 25 Hz, according to Eq. (3) [16].


**Table 2.** Specification of three-phase LSPMSM

Y connected, and the current nominal value is 1.28 A. The LSPMSM is coupled to torque/speed sensor in order to measure the torque value in different operation conditions. On the other side, a mechanical load is provided by a DC-excited magnetic powder brake (MPB) coupled to torque/speed sensor. The specific load torque level could be furnished to the motor shaft by controlling the input dc voltage of MPB. This system is used to sample the stator current noninvasively when the motor is operated in the steady-state condition. Notably, only one phasecurrent signal is required to be recorded for the detection process in this study. The recorded

The method proposed in this study for the creation of eccentricity fault in LSPMSM, by changing the original bearings of motor with a new set of bearing with larger inner diameter and smaller outer diameter, results in the creation of free space between the shaft and bearings and also between the bearings and the housing of end shields. Static eccentricity is created by fixing concentric inner rings between the new bearings and shaft on both ends of LSPMSM and non-concentric outer rings between the new bearings and housings of both end shields. The aforementioned strategy is used to create 33% and 50% static eccentricity in the motor

The current spectrum is stored with the sampling frequency ( *f* <sup>S</sup>) of 5 kHz over a total sampling period of 6.5 s, which allows the analysis of the signals with a minimum frequency of 0.15 Hz. Daubechies-24 (db24) is used as the mother wavelet in discrete wavelet transform (DWT) analyses. Since the four-pole, three-phase LSPMSM is considered, the characteristic frequency component associated with static eccentricity is located at 25 Hz, according to Eq. (3) [16].

signals are analysed by a computer-based signal processing program.

96 Wavelet Transform and Some of Its Real-World Applications

**Figure 4.** Experimental test rig

discussed in the case study.

**5.2. Method**

The number of decomposition levels (*l* d) can be determined using Eq. (16) which is *l* <sup>d</sup> =7 in this case.

$$l\_d = \frac{\log\left(f\_{\rm S} / f\_{\rm static}\right)}{\log\left(2\right)}\tag{16}$$

The frequency bands of wavelet signals are summarized in Table 3. Energies of the detail coefficient *E*(*Dj* ) are calculated using the following formulas [45]:

$$E\left(D\_{\cdot}\right) = \sqrt{\frac{1}{N\_1} \sum\_{i=1}^{N\_1} \left(D\_{\cdot}\right)^2 \left[i\right]}\tag{17}$$

where *j* =1, 2, ..., *l* d and *N*<sup>l</sup> is the data length of the decomposition level.


**Table 3.** The frequency bands of wavelet signals

#### **5.3. Results and discussion**

Figure 5 shows the stator current signal (original signal) and *D*5, *D*6, and *D*<sup>7</sup> are the detail signals obtained by db24 at level 7 for fault-free LSPMSM. The fault-related components ( *f* static) are visible at 25 Hz, which confirm the productivity of *D*<sup>7</sup> signal for the detection of static eccen‐ tricity. The original and detail signals of LSPMSM with 33% and 50% static eccentricity are indicated in Figures 6 and 7, respectively.

**Figure 5.** DWT analysis of current signal of fault-free LSPMSM

A comparison between Figures 5, 6, and 7 shows that the signals of *D*7 are clear from any distortion in a fault-free motor while the high distortions are manifested in *D*<sup>7</sup> in the presence of static eccentricity that demonstrates the faulty condition of LSPMSM. The source of these distortions is due to the increase in the amplitudes of fault-related frequency components based on Eq. (3).

tricity. The original and detail signals of LSPMSM with 33% and 50% static eccentricity are

indicated in Figures 6 and 7, respectively.

98 Wavelet Transform and Some of Its Real-World Applications

**Figure 5.** DWT analysis of current signal of fault-free LSPMSM

**Figure 6.** DWT analysis of current signal of LSPMSM under 33% static eccentricity

**Figure 7.** DWT analysis of current signal of LSPMSM under 50% static eccentricity

An effective static eccentricity detection index is introduced for three-phase LSPMSM based on the energy of *D*7 for the stator current signal. The proposed index is examined for fault-free and eccentric LSPMSM with 33% and 50% static eccentricity as shown in Figure 8. The energy variation of *D*7 (index) for stator current signal using db24 is provided in Table 4.

**Figure 8.** Static eccentricity severity versus proposed index (energy of *D*7)


**Table 4.** Evaluation of proposed index due to fault degree

#### **5.4. Conclusion**

Discrete wavelet transform is employed to analyse the stator current signal of three-phase LSPMSM in order to propose an effective index for static eccentricity fault detection. The energy of detail signal (*D*7) is introduced as eccentricity index. The achieved results confirm the productivity of the proposed method for the motor discussed in the case study.

#### **6. Case study 2**

The detection of broken rotor bar in three-phase squirrel cage induction motor using motor current signature analysis is studied. A detailed description of experimental test rig used in this study and the method used for signal measurement and analysis is explained in the following.

#### **6.1. Experimental set-up**

**Figure 7.** DWT analysis of current signal of LSPMSM under 50% static eccentricity

100 Wavelet Transform and Some of Its Real-World Applications

An effective static eccentricity detection index is introduced for three-phase LSPMSM based on the energy of *D*7 for the stator current signal. The proposed index is examined for fault-free and eccentric LSPMSM with 33% and 50% static eccentricity as shown in Figure 8. The energy

variation of *D*7 (index) for stator current signal using db24 is provided in Table 4.

Figure 9 illustrates the experimental test rig used in this study. Table 5 presents the parameters of the three-phase squirrel cage induction motor used for both fault-free and faulty motors. The faulty motor is with three broken rotor bars. The motor is directly fed by the grid power supply, while the stator windings are Y connected and the current nominal value is 2.2 A. In order to measure the torque and speed value of the squirrel cage induction motor in different operation conditions, a torque/speed sensor is coupled to it. A generator is used as a load and the specific load torque level can be furnished to the motor shaft by controlling the resistor connected to the generator. Recall that only one phase current signal is required to be recorded for the detection process in this study. The recorded signals are then analysed by a computerbased signal processing program.

**Figure 9.** Experimental set-up


**Table 5.** Specification of a three-phase squirrel cage induction motor

#### **6.2. Method**

The architecture of the proposed system for broken rotor bar detection is shown in Figure 10, and the procedure used in this study is as follows: First, to force a real bar breakage in the rotor, a hole is drilled artificially in it. The original stator current was recorded from a threephase induction motor. The stator current is sampled at 20 kHz lasting four seconds for both fault-free and faulty motors (three-rotor bar breakage) at 80% full load. The measured current signals are then decomposed using wavelet packet transform with two different purposes: One as a pre-processing of signal for FFT analysis and the frequency of ((1*–*2*s*)*f*S) obtained is used as a fault feature for broken rotor bar detection. The other purpose of using WPT is for feature extraction, where some statistical features determined by wavelet packet coefficients are used for broken rotor bar detection. For both purposes, Daubechies-44 (db44) is applied as a mother wavelet in 12 levels of decomposition. To extract the fault-related feature, those nodes are taken that involve fault frequency (*f*BRB). Figure 11 shows the process explained above, called the wavelet packet tree. In this work, the signal energy, root mean square (RMS), and kurtosis are obtained as selected features for the diagnosis of the broken rotor bar.

operation conditions, a torque/speed sensor is coupled to it. A generator is used as a load and the specific load torque level can be furnished to the motor shaft by controlling the resistor connected to the generator. Recall that only one phase current signal is required to be recorded for the detection process in this study. The recorded signals are then analysed by a computer-

Rated output power (HP) 1 Rated voltage (V) 415 Rated frequency (Hz) 50 Number of poles 6 Rated speed (RPM) 1000 Connection Y Number of rotor bars 28

The architecture of the proposed system for broken rotor bar detection is shown in Figure 10, and the procedure used in this study is as follows: First, to force a real bar breakage in the rotor, a hole is drilled artificially in it. The original stator current was recorded from a threephase induction motor. The stator current is sampled at 20 kHz lasting four seconds for both fault-free and faulty motors (three-rotor bar breakage) at 80% full load. The measured current signals are then decomposed using wavelet packet transform with two different purposes: One as a pre-processing of signal for FFT analysis and the frequency of ((1*–*2*s*)*f*S) obtained is used as a fault feature for broken rotor bar detection. The other purpose of using WPT is for feature

**Table 5.** Specification of a three-phase squirrel cage induction motor

based signal processing program.

102 Wavelet Transform and Some of Its Real-World Applications

**Figure 9.** Experimental set-up

**6.2. Method**

**Figure 10.** The architecture of the proposed system for broken rotor bar detection

**Figure 11.** Approximations and details in wavelet packet decomposition

#### **6.3. Results and discussion**

In order to obtain the differences between fault-free and faulty conditions under 80% full-load conditions, WPD was used for the feature extraction. The WPD gives distinguishable signa‐ tures from stator current signal in a specific frequency band. After WPD of the current signal, two procedures for failure feature extraction using WPD are used (Figure 10). One procedure includes using FFT for the determination of amplitude of fault frequency and the other includes the statistical analysis of coefficients extracted by WPD.

The amplitude of fault frequency in the current spectrum for fault-free motors and for motors with three broken bars achieved in the first procedure is presented in Table 6. The results indicate that the amplitude of harmonic components ((1*–*2*s*)*f*S) in both nodes, presented in Table 6, increase the faulty condition. However, the degree of increase is not significant, and it cannot be used to differentiate the conditions.


**Table 6.** Amplitudes of harmonic components for fault-free and faulty motors

In the second procedure, three statistical parameters including RMS, kurtosis and energy are calculated using the statistical analysis of coefficients determined by WPD of current signal. Table 7 presents these statistical parameters in three different nodes [10, 6], [11, 13] and [12, 26]. These parameters are compared to define the most appropriate frequency band that represents the frequency components from the broken rotor bar. According to Table 7, the nodes [11, 3] (46.39–48.83 Hz) in wavelet packet tree are the most dominant bands that can differentiate between fault-free and faulty motors under full load.


**Table 7.** Statistical features for fault-free and faulty motors

#### **6.4. Conclusions**

**6.3. Results and discussion**

104 Wavelet Transform and Some of Its Real-World Applications

*f*BRB *=* (1–2*s*)*fS =* 47.604 Hz

the statistical analysis of coefficients extracted by WPD.

**Node Frequency Band Amplitude (1-2s)fs**

**Table 6.** Amplitudes of harmonic components for fault-free and faulty motors

differentiate between fault-free and faulty motors under full load.

**Table 7.** Statistical features for fault-free and faulty motors

**Feature** Condition [10, 6] [11, 13] [12, 26] **RMS** Fault-free 11.82 15.93 40.36

**Kurtosis** Fault-free 2.94 2.97 2.16

**Energy** Fault-free 14,816 24,375 172,684

[10, 6] (39.06–48.83) 0.43 0.43 [11, 13] (46.39–48.83) 0.42 0.42

it cannot be used to differentiate the conditions.

In order to obtain the differences between fault-free and faulty conditions under 80% full-load conditions, WPD was used for the feature extraction. The WPD gives distinguishable signa‐ tures from stator current signal in a specific frequency band. After WPD of the current signal, two procedures for failure feature extraction using WPD are used (Figure 10). One procedure includes using FFT for the determination of amplitude of fault frequency and the other includes

The amplitude of fault frequency in the current spectrum for fault-free motors and for motors with three broken bars achieved in the first procedure is presented in Table 6. The results indicate that the amplitude of harmonic components ((1*–*2*s*)*f*S) in both nodes, presented in Table 6, increase the faulty condition. However, the degree of increase is not significant, and

In the second procedure, three statistical parameters including RMS, kurtosis and energy are calculated using the statistical analysis of coefficients determined by WPD of current signal. Table 7 presents these statistical parameters in three different nodes [10, 6], [11, 13] and [12, 26]. These parameters are compared to define the most appropriate frequency band that represents the frequency components from the broken rotor bar. According to Table 7, the nodes [11, 3] (46.39–48.83 Hz) in wavelet packet tree are the most dominant bands that can

Faulty 12.96 17.78 22.06

Faulty 3.3 3.61 3.09

Faulty 17,814 30,343 44,277

**Fault-free Faulty**

This case study proposes a feature extraction system for broken rotor bar detection using wavelet packet coefficients of the stator current. It is shown that in a faulty case, the amplitude in specific side bands increases and dominant features of signals can be extracted for fault diagnostics. The results of this study indicate that the energy, kurtosis and RMS value of WPD coefficients are the appropriate features for detecting broken rotor bar in particular bands.

#### **Acknowledgements**

The authors would like to express their gratitude to the Ministry of Education, Malaysia for their financial support through grant number FRGS-5524356 and Universiti Putra Malaysia for the facilities provided during this research work.

#### **Author details**

Mohammad Rezazadeh Mehrjou1,2, Norman Mariun1,2, Mahdi Karami1,2, Samsul Bahari Mohd. Noor2 , Sahar Zolfaghari2 , Norhisam Misron2 , Mohd Zainal Abidin Ab. Kadir2 , Mohd. Amran Mohd. Radzi1,2 and Mohammad Hamiruce Marhaban2

\*Address all correspondence to: Mehrjou.mo@gmail.com

1 Centre for Advanced Power and Energy Research (CAPER), Faculty of Engineering, Universiti Putra Malaysia, Serdang, Malaysia

2 Department of Electrical and Electronic Engineering, Faculty of Engineering, Universiti Putra Malaysia, Serdang, Malaysia

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Additional information is available at the end of the chapter

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Owing to the relevance and severity of damages caused by earthquakes (EQs), the development and application of new methods for seismic activity detection that offer an efficient and reliable diagnosis in terms of processing and perform‐ ance are still demanding tasks. In this work, the application of the Empirical Wavelet Transform (EWT) for seismic detection in ultra-low-frequency (ULF) geomagnetic signals is presented. For this, several ULF signals associated to seismic activities and random calm periods are analysed. These signals have been obtained through a tri-axial fluxgate magnetometer at the Juriquilla station localized in Queretaro, Mexico, longitude -100.45° N and latitude 20.70°E. In or‐ der to show the advantages of the proposal, a comparison with the discrete wavelet transform (DWT) is presented. The results shown a better detection ca‐ pability of seismic signals before, during, and after the main shock than the ones obtained by the DWT, which makes the proposal a more suitable and relia‐ ble tool for this task. Finally, a fuzzy logic (FL)-based system for automatic di‐ agnosis using the variance of the EWT outputs for the tri-axial fluxgate magnetometer signals is also proposed.

**Keywords:** Empirical Wavelet Transform, Time-Frequency Analysis, Earthquake in‐ teraction, forecasting, prediction

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **1. Introduction**

Among the natural disasters, the EQs have attracted the interest of many researchers around the world due to the huge amount of human, material, and economic losses [1]. They have been focused on finding pre-seismic precursors for prediction and forecasting tasks [2-6]. In this regard, different electromagnetic phenomena (EP) encompassing a large frequency range, being the ultra-low-frequency (ULF) range one of the most promising, have been associated with EQs because they typically occur during, but sometimes prior to seismic activity [7-14] Although many detection methods have been proposed, the relevance and severity of EQs damages demand still more efficient and reliable diagnosis methods.

Techniques and methods such as polarization or spectral density ratio analysis [15-16], transfer function analysis [17], fractal analysis [18-22], singular value decomposition [23], principal component analysis [16, 21], and the discrete wavelet transform (DWT) [2, 24], among others have been proposed to analyse the ULF geomagnetic signals associated to EQs. Yet, despite showing promising results, the inherent characteristics of the ULF geomagnetic signals such as high noise levels, weak amplitude, and interferences from other sources due to the distance between the epicenter and sensor, among others may compromise the performance and reliability of the analysis. From this point of view, the development and application of new detection methods to make a more efficient and reliable diagnosis in terms of processing and performance are still interesting research fields.

In this chapter, the application of the empirical wavelet transform (EWT) to ULF signals for detecting seismic precursor anomalies is presented. Besides, a comparison with the DWT is carried out in order to show the EWT advantages. Moreover, a fuzzy logic (FL) system for automatic diagnosis using the variance of the EWT results is proposed. For this, three ULF signals associated to seismic activities and random calm periods are analysed. The obtained results show the usefulness and effectiveness of the proposed methodology, making it a suitable and reliable tool to detect ULF anomalies.

#### **2. ULF geomagnetic data**

In order to investigate the relationships between ULF geomagnetic signals and pre-seismic anomalies, ULF geomagnetic data from Juriquilla seismic station, located in Queretaro, Mexico, with geographic coordinates: longitude -100.45° N and latitude 20.70° E, are used. The ULF geomagnetic signals are monitored by means of a fluxgate magnetometer. It allows monitoring three mutually orthogonal components of the magnetic field, two horizontal (Mx: North-South and My: East-West) components and a vertical component (Mz). The three geomagnetic components are measured using a sampling frequency of 1 Hz to obtain 65,000 samples during a time window of 18 hrs, which comprise 9 hours before the main shock and 9 hrs after it. In this research, three recent seismic events with magnitude greater than 6.0 are analysed. Further, for comparison purposes, random analyses during periods of seismic calm are used. Table 1 summarize the characteristics of the studied EQs.

To discriminate the geomagnetic activity of the magnetosphere because of the solar activity and cultural noise, the analysed EQs data are compared with the geomagnetic activity expressed by Dst index (http://wdc.kugi.kyoto-u.ac.jp/dstdir/), where those indices apparently had no correlation with EQs variation.


Note: The Year / month / day / hour / minute are the exact time of the EQ (Local Time); Latitude and Longitude are the geographic coordinates of the epicenter, Magnitude and Depth are the EQ measures, Distance is the distance between the epicenter and Juriquilla station, and p is the radius of the EQ preparation.

**Table 1.** Characteristics of the Earthquakes occurred in Mexico during 2009–2010. Their magnitudes are presented in bold (Catalogue of National Seismological Service, Mexico)

#### **3. Wavelet transform**

**1. Introduction**

112 Wavelet Transform and Some of Its Real-World Applications

Among the natural disasters, the EQs have attracted the interest of many researchers around the world due to the huge amount of human, material, and economic losses [1]. They have been focused on finding pre-seismic precursors for prediction and forecasting tasks [2-6]. In this regard, different electromagnetic phenomena (EP) encompassing a large frequency range, being the ultra-low-frequency (ULF) range one of the most promising, have been associated with EQs because they typically occur during, but sometimes prior to seismic activity [7-14] Although many detection methods have been proposed, the relevance and severity of EQs

Techniques and methods such as polarization or spectral density ratio analysis [15-16], transfer function analysis [17], fractal analysis [18-22], singular value decomposition [23], principal component analysis [16, 21], and the discrete wavelet transform (DWT) [2, 24], among others have been proposed to analyse the ULF geomagnetic signals associated to EQs. Yet, despite showing promising results, the inherent characteristics of the ULF geomagnetic signals such as high noise levels, weak amplitude, and interferences from other sources due to the distance between the epicenter and sensor, among others may compromise the performance and reliability of the analysis. From this point of view, the development and application of new detection methods to make a more efficient and reliable diagnosis in terms of processing and

In this chapter, the application of the empirical wavelet transform (EWT) to ULF signals for detecting seismic precursor anomalies is presented. Besides, a comparison with the DWT is carried out in order to show the EWT advantages. Moreover, a fuzzy logic (FL) system for automatic diagnosis using the variance of the EWT results is proposed. For this, three ULF signals associated to seismic activities and random calm periods are analysed. The obtained results show the usefulness and effectiveness of the proposed methodology, making it a

In order to investigate the relationships between ULF geomagnetic signals and pre-seismic anomalies, ULF geomagnetic data from Juriquilla seismic station, located in Queretaro, Mexico, with geographic coordinates: longitude -100.45° N and latitude 20.70° E, are used. The ULF geomagnetic signals are monitored by means of a fluxgate magnetometer. It allows monitoring three mutually orthogonal components of the magnetic field, two horizontal (Mx: North-South and My: East-West) components and a vertical component (Mz). The three geomagnetic components are measured using a sampling frequency of 1 Hz to obtain 65,000 samples during a time window of 18 hrs, which comprise 9 hours before the main shock and 9 hrs after it. In this research, three recent seismic events with magnitude greater than 6.0 are analysed. Further, for comparison purposes, random analyses during periods of seismic calm

damages demand still more efficient and reliable diagnosis methods.

performance are still interesting research fields.

suitable and reliable tool to detect ULF anomalies.

are used. Table 1 summarize the characteristics of the studied EQs.

**2. ULF geomagnetic data**

This section presents the theoretical background of the Discrete Wavelet transform and the Empirical Wavelet Transform used for the analysis of ULF signals.

#### **3.1. Discrete wavelet transform**

Discrete Wavelet Transform (DWT) is a useful method for analysis of non-stationary, no linear and transient signals because it decomposes the time series signal into multiple time-frequency levels retaining the characteristics of the analysed signal [2]. DWT is defined by Eq. (1), where *x*(n) and *h*(n) denote the discrete signal and the wavelet basis function, respectively, of the total number *N* of samples contained in the signal *x*(n). *j* and *k* represent the time scaling, and the shifting of the discrete wavelet function, respectively.

$$DWT\_{j,k} = \sum\_{N} \mathbf{x}(n) \overline{h\_{j,k}(n)} \tag{1}$$

The DWT is calculated using a set of low- and high-pass filters bank called approximations (ACL) and details (DCL) into desired levels *L* (Mallat algorithm), respectively, as shown in Figure 1. Based on the Mallat algorithm, the approximation obtained from the first level is split into a new decomposition and approximation and this process is repeated [25]. Once the discrete time signal *x*(n) has been decomposed into the desired levels, the signal is recon‐ structed by applying the decomposition process in an inverse way, which is known as the inverse discrete wavelet transform (IDWT).

**Figure 1.** DWT basis construction.

According to the Mallat algorithm, the frequency band for the approximations *ACL* and decompositions *DCL DC* are given by Eq. (2) and Eq. (3), respectively, where *Fs* represents the sampling frequency of the signal.

$$\mathbf{AC}\_{L} \Rightarrow \left[\mathbf{0}, \frac{F\_{s}}{2^{L\*1}}\right] \tag{2}$$

$$\text{DC}\_{L} \Rightarrow \left[\frac{F\_s}{2^{L+1}}, \frac{F\_s}{2^L}\right] \tag{3}$$

Different types of wavelet mother function have been proposed to analyse ULF signals in order to find anomalies related with EQs such as Daubechies, Haar, Morlet, Symlets, Coiflets, and Meyer (Figure 2). However, it has been demonstrated that the most effective to analyse ULF signals is the Daubechies mother function [2, 24, 26]. For this reason, Daubechies as mother wavelet is used in this work.

#### **3.2. Empirical wavelet transform**

EWT is a new adaptive wavelet transform capable of decomposing a time series signal *x*(t) into adaptive time-frequency sub-bands according to its contained frequency information [27]. This advantage allows generating narrow time-frequency sub-bands, unlike the DWT where the calculated time-frequency sub-bands depend on sampling frequency of the time signal. To provide an adaptive wavelet with respect to the analysed signal, the segmentation of the signal is an important step in the EWT. It can be performed either manually or by means of the Fourier spectrum. If the signal is segmented manually, the boundaries of the wavelet filters can be user selected as contiguous segments. On the other hand, using the Fourier spectrum, first, the local maxima of the Fourier spectrum *x*(*ω*) *x* are calculated. Next, the boundaries *ω<sup>i</sup> ω* of each segment are defined as the center between two consecutive maxima. Thus, the Fourier support [0, π] is segmented *N* into contiguous sets or frequency bands. In both segmentations, each frequency band is indicated by Λ<sup>n</sup> = ωn−1, ω<sup>n</sup> Λn=[ω and satisfy, as shown in Figure. 3. A

Empirical Wavelet Transform-based Detection of Anomalies in ULF Geomagnetic Signals Associated to… http://dx.doi.org/10.5772/61163 115

**Figure 2.** Wavelets used in ULF signals.

**Figure 1.** DWT basis construction.

sampling frequency of the signal.

114 Wavelet Transform and Some of Its Real-World Applications

wavelet is used in this work.

**3.2. Empirical wavelet transform**

According to the Mallat algorithm, the frequency band for the approximations *ACL* and decompositions *DCL DC* are given by Eq. (2) and Eq. (3), respectively, where *Fs* represents the

> <sup>1</sup> 0, <sup>2</sup> *s*

<sup>1</sup> , 2 2 *s s L L L F F DC* <sup>+</sup>

Different types of wavelet mother function have been proposed to analyse ULF signals in order to find anomalies related with EQs such as Daubechies, Haar, Morlet, Symlets, Coiflets, and Meyer (Figure 2). However, it has been demonstrated that the most effective to analyse ULF signals is the Daubechies mother function [2, 24, 26]. For this reason, Daubechies as mother

EWT is a new adaptive wavelet transform capable of decomposing a time series signal *x*(t) into adaptive time-frequency sub-bands according to its contained frequency information [27]. This advantage allows generating narrow time-frequency sub-bands, unlike the DWT where the calculated time-frequency sub-bands depend on sampling frequency of the time signal. To provide an adaptive wavelet with respect to the analysed signal, the segmentation of the signal is an important step in the EWT. It can be performed either manually or by means of the Fourier spectrum. If the signal is segmented manually, the boundaries of the wavelet filters can be user selected as contiguous segments. On the other hand, using the Fourier spectrum, first, the local maxima of the Fourier spectrum *x*(*ω*) *x* are calculated. Next, the boundaries *ω<sup>i</sup> ω* of each segment are defined as the center between two consecutive maxima. Thus, the Fourier support [0, π] is segmented *N* into contiguous sets or frequency bands. In both segmentations, each frequency band is indicated by Λ<sup>n</sup> = ωn−1, ω<sup>n</sup> Λn=[ω and satisfy, as shown in Figure. 3. A

é ù Þ ê ú

ë û (2)

ë û (3)

é ù Þ ê ú

*L L <sup>F</sup> AC* <sup>+</sup> transition phase of width 2*τn* 2τ is defined to obtain a tight frame around each ω<sup>n</sup> *ω*. A more detailed selection of *τn* τ is presented in [27]. Observing Figure. 3, the empirical wavelets are defined by one low-pass represented by LPF *ϕn*(*ω*) and *N* −1 *N* band-pass *ψn*(*ω*) filters represented by BPF corresponding to the approximation and details components, respectively, on each Λn Λn.

**Figure 3.** EWT basis construction.

Following the idea used in deriving the Meyer's wavelet, [27] defines the empirical scaling function to estimate the low-pass wavelet filter coefficients according to following Equation:

$$\phi\_n(o) = \begin{cases} 1 & \text{if } \left| oo \right| \le o\_n - \tau\_n \\ \cos \left[ \frac{\pi}{2} \beta \left( \frac{1}{2\tau\_n} \left( \left| oo \right| - o\_n + \tau\_n \right) \right) \right] \text{if } oo\_n - \tau\_n \le \left| oo \right| \le o\_n + \tau\_n \\ 0 & \text{otherwise} \end{cases} \tag{4}$$

And an empirical wavelet function to build the *N-*1 band-pass filters as:

$$\nu\_n(\boldsymbol{\alpha}) = \begin{cases} 1 & \text{if } \boldsymbol{\alpha}\_n + \boldsymbol{\tau}\_n \le |\boldsymbol{\alpha}| \le \boldsymbol{\alpha}\_{n+1} - \boldsymbol{\tau}\_{n+1} \\\\ \cos\left[\frac{\pi}{2}\beta \left(\frac{1}{2\boldsymbol{\tau}\_{n+1}} \left(|\boldsymbol{\alpha}| - \boldsymbol{\alpha}\_{n+1} + \boldsymbol{\tau}\_{n+1}\right)\right) \text{if } \boldsymbol{\alpha}\_{n+1} - \boldsymbol{\tau}\_{n+1} \le |\boldsymbol{\alpha}| \le \boldsymbol{\alpha}\_{n+1} + \boldsymbol{\tau}\_{n+1} \\\\ \sin\left[\frac{\pi}{2}\beta \left(\frac{1}{2\boldsymbol{\tau}\_{n}} (|\boldsymbol{\alpha}| - \boldsymbol{\alpha}\_{n} + \boldsymbol{\tau}\_{n}\right)\right) \text{if } \boldsymbol{\alpha}\_{n} + \boldsymbol{\tau}\_{n} \le |\boldsymbol{\alpha}| \le \boldsymbol{\alpha}\_{n} + \boldsymbol{\tau}\_{n} \\\\ 0 & \text{otherwise} \end{cases} \tag{5}$$

where *β*(x) is a polynomial function taken as *β*(*x*)= *x* <sup>4</sup> (35−85*x* + 70*x* <sup>2</sup> −20*x* <sup>3</sup> )

After having built the wavelet filters, the signal *x*(t) is decomposed into different frequency bands through empirical wavelet transform defined by

$$\mathcal{W}\_f^\epsilon \begin{pmatrix} \mathfrak{u}, t \end{pmatrix} = F^{-1} \left( \mathfrak{x}(o\sigma) \mathcal{Y}\_n \begin{pmatrix} o \end{pmatrix} \right) \tag{6}$$

$$\mathcal{W}\_f^\epsilon \begin{pmatrix} 0, t \end{pmatrix} = F^{-1} \left( \mathbf{x}(\alpha) \phi\_u \begin{pmatrix} \alpha \\ \end{pmatrix} \right) \tag{7}$$

where the details *Wf* (*n*, *t*) and approximation *Wf* (0, *t*) coefficients are obtained by the inner products of the signal with the empirical wavelets low-pass and band-pass filters, respectively, and*F*-1is the inverse Fourier transform.

#### **4. EWT and fuzzy logic system**

This section presents the proposed methodology. It consists of the ULF geomagnetic signals analysis through the EWT, then a statistical parameter based on the variance is applied and, finally, an automatic diagnosis by means of a FL system is computed as shown in Figure. 4.

**Figure 4.** Proposed methodology.

#### **4.1. EWT analysis**

And an empirical wavelet function to build the *N-*1 band-pass filters as:

1

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1 2 2

*n*

t

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ò

(*n*, *t*) and approximation *Wf*

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where *β*(x) is a polynomial function taken as *β*(*x*)= *x* <sup>4</sup>

bands through empirical wavelet transform defined by

p b

116 Wavelet Transform and Some of Its Real-World Applications

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wt

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products of the signal with the empirical wavelets low-pass and band-pass filters, respectively,

This section presents the proposed methodology. It consists of the ULF geomagnetic signals analysis through the EWT, then a statistical parameter based on the variance is applied and, finally, an automatic diagnosis by means of a FL system is computed as shown in Figure. 4.

 t

<sup>ï</sup> é ù æ ö <sup>ï</sup> ê ú ç ÷ - + +£ £ + <sup>ï</sup> ë û è ø <sup>ï</sup>

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wy w

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 ww

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**Figure 4.** Proposed methodology.

and*F*-1is the inverse Fourier transform.

**4. EWT and fuzzy logic system**

î

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y w Generally, the pre-seismic anomalies are too much weak to be detected by the Fourier transform (Chavez et al., 2010); hence, the frequency bands are selected manually using the EWT. Several experimental using both algorithms are carried out to estimate the best frequency band of the ULF geomagnetic signal to detect anomalies associated to the EQs. After the experimental runs, it is found that for the EWT algorithm the frequency band from 0.0470 to 0.0781 Hz and for the DWT algorithm the frequency band or third decomposition from 0.0625 to 0.125 Hz with a Daubechies wavelet of order 5 generate the best results, enhancing corre‐ lation with associated seismic anomalies events. Figure. 5 presents the obtained results for the EWT and the DWT, where both the seismic calm signal (left-side plots) and the seismic activity signal (right-side plots), which corresponds to event 1 (Table 1), can be observed. Figure.5(a) shows that in both analysis, the seismic calm period do not present significant spikes over time, indicating the absence of seismic activity. On the other hand, observing the results shown in Figure. 5(b), both time-frequency analysis can detect the occurrence of peaks prior to seismic (Pre-seismic event zone) and another peaks after the main shock (Post-seismic event zone). These magnetic perturbations occur about 8hrs before the main shock and about 2 hrs after it. Open circles remark perturbations in the signal.. But, it is noticeable that, using the EWT method, it is better noticeable of the pre-seismic and post-seismic anomalies.

**Figure 5.** Comparison between the EWT time-frequency analysis and DWT time-frequency analysis; for (a) seismic calm, and (b) with seismic activity.

In order to evaluate the significance of these results, a complementary statistical analysis based on the variance of the EWT and DWT results is computed to measure the fluctuations between seismic activity and seismic calm period as follows:

$$V = \frac{1}{N} \sum\_{n=1}^{N} \left[ \mathbf{x}(n) - \overline{\mathbf{x}} \right] \tag{8}$$

where *V* is the variance of each data window at the frequency band, *N* is the total of data analysed for the region of interest, *x*(n) is the input sequence, and *x*¯ is the mean value of *x*(n). Figures. 6(a) 6(b) show the variance (*V*) results obtained by the EWT and the DWT method, respectively, for seismic activity and seismic calm period. The results correspond to running data windows each 1000 samples. It is observed that the EWT method presents a better performance to detect seismic anomalies than DWT method. Hence, it can be established that the EWT analysis allows the observation of ULF signal perturbations with low amplitude and embedded in high level of noise.

**Figure 6.** Variance (V) of the seismic activity and seismic calm period using: (a) EWT and (b) DWT.

#### **4.2. Study cases**

After showing in the previous section that EWT improves the correlation between the seismic event and the ULF electromagnetic signal, the proposed EWT time-frequency analysis is applied to seismic calm and three seismic events with different geographical location. Figure. 7 shows the EWT time-frequency analysis for seismic calm period and the 3 seismic events. The three components of the magnetic field, Mx, My, and Mz, are analysed as shown in Figure. 7. The main shock position is indicated by an arrow and two circles. It shows the pre-seismic and post-seismic anomalies associated with the EQs. The EWT results for seismic calm period do not present significant spikes over time, indicating the absence of seismic activity, as shown in Figure. 7(a-c); unlike the signals with seismic activity where several spikes appear before and after the main shock. All the analysed signals comprise 9 hrs before and 9 hrs after each seismic event, considering the time zero as the specific time of the occurrence of the EQ.

Similar to previous section, to evaluate the significance of the results, a complementary statistical analysis based on the variance of the EWT results is computed to measure the fluctuations between seismic activity and seismic calm period. Figure. 8 shows the variance V results obtained for three analysed seismic events as well as for their three components (Mx, My, and Mz). The results correspond to running data windows each 1000 samples. According

Empirical Wavelet Transform-based Detection of Anomalies in ULF Geomagnetic Signals Associated to… http://dx.doi.org/10.5772/61163 119

( )

where *V* is the variance of each data window at the frequency band, *N* is the total of data analysed for the region of interest, *x*(n) is the input sequence, and *x*¯ is the mean value of *x*(n). Figures. 6(a) 6(b) show the variance (*V*) results obtained by the EWT and the DWT method, respectively, for seismic activity and seismic calm period. The results correspond to running data windows each 1000 samples. It is observed that the EWT method presents a better performance to detect seismic anomalies than DWT method. Hence, it can be established that the EWT analysis allows the observation of ULF signal perturbations with low amplitude and

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1

1 *<sup>N</sup> n V xn x N* <sup>=</sup>

**Figure 6.** Variance (V) of the seismic activity and seismic calm period using: (a) EWT and (b) DWT.

After showing in the previous section that EWT improves the correlation between the seismic event and the ULF electromagnetic signal, the proposed EWT time-frequency analysis is applied to seismic calm and three seismic events with different geographical location. Figure. 7 shows the EWT time-frequency analysis for seismic calm period and the 3 seismic events. The three components of the magnetic field, Mx, My, and Mz, are analysed as shown in Figure. 7. The main shock position is indicated by an arrow and two circles. It shows the pre-seismic and post-seismic anomalies associated with the EQs. The EWT results for seismic calm period do not present significant spikes over time, indicating the absence of seismic activity, as shown in Figure. 7(a-c); unlike the signals with seismic activity where several spikes appear before and after the main shock. All the analysed signals comprise 9 hrs before and 9 hrs after each seismic event, considering the time zero as the specific time of the occurrence of the EQ.

Similar to previous section, to evaluate the significance of the results, a complementary statistical analysis based on the variance of the EWT results is computed to measure the fluctuations between seismic activity and seismic calm period. Figure. 8 shows the variance V results obtained for three analysed seismic events as well as for their three components (Mx, My, and Mz). The results correspond to running data windows each 1000 samples. According

embedded in high level of noise.

118 Wavelet Transform and Some of Its Real-World Applications

**4.2. Study cases**

**Figure 7.** EWT analysis of the three geomagnetic components: Mx, My and Mz, for a seismic calm (a-c), and with seis‐ mic activity: (d) to (f) EQ1, (g) to (i) EQ2, and (j) to (l) EQ3.

to the obtained results in Figure.8(a-c), there are important variations of V for the three components that could be associated with occurrence of the EQs. As observed, V increases before, during and, after the seismic event on the three components: Mx (a), My (b), and Mz (c). Observing Figure. 8, the Mx geomagnetic component presents an important variance in three different ranges: from 8 to 5 hrs before seismic event, between 2hr before and 2hr after the main shock, and from 3 to 7 hrs after the main EQs (post-seismic zone). The My component shows variance in different ranges, from 8 to 6 hrs before seismic event, between 2 hrs before and 2 hrs after main shock, and from 3 to 8 hrs after the main EQs. Finally, the Mz geomagnetic component also presents variance in three different ranges: from 8 to 6 hrs before seismic event, 2 hrs before and 2 hrs after main shock, and from 4 to 8 hrs after seismic event. In summary, these results show that the EWT is adequate to find electro-magnetic seismic precursors related to the variance magnitude.

**Figure 8.** Variance of the EWT for the three geomagnetic components: (a) Mx, (b) My, and (c) Mz.

#### **4.3. Fuzzy logic-based system**

Once the variance of the EWT signals is computed, a FL-based system is used for automatically diagnosing the severity of the ULF geomagnetic variations associated to seismic events. A FL system represents a group of rules for reasoning under uncertainty in an imprecise or fuzzy manner. It is usually used when a mathematical model of a process does not exist or does exist but is too difficult to encode and too complex to be evaluated fast enough for real time operation. Besides, it can use several sources of information in order to take a decision according to a particular objective.

The designed and implemented FL system to perform the diagnosis process is a Mamdanitype fuzzy inference system with two inputs, one output, and 16 rules. The system uses Max– Min composition, and the centroid of area method for defuzzification. The inputs are the variance of EWT results for the signals Mx and My, the Mz signal is not considered since it presents a low difference between seismic activity and calm period. These inputs are parti‐ tioned into four trapezoidal membership function sets, as shown in Figures. 9(a) and 9(b). They are labelled as NV (normal variance), LV (low variance), MV (medium variance), and HV (high variance). The output is also divided into four trapezoidal membership functions as shown in Figure. 9(c), their labels are NF (normal fluctuations), LF (low fluctuations), MF (medium fluctuations), and HF (high fluctuations). The crisp output of the Mamdani FL system can assume values between 0.5 and 4.5, where normal variations = 1, low variations = 2, medium variations = 3, and high variations = 4. The parameters of membership functions are determined according to the interpretation of the variance results by the authors. The set of rules that classifies the inputs variance is show in Table 2; there, one rule can be read as follows if (variance Mx is NV and variance My is NV) (light gray) then the geomagnetic fluctuations magnitude is NF (dark gray), and so on.


**Table 2.** Rules table for the proposed FL

**Figure 9.** Membership functions: (a) Input Mx variance, (b) Input My variance, and (c) Output diagnosis.

The FL output for a calm period signal is shown in Figure. 10(a), where most of the results are Normal and only a few data indicate Low ULF geomagnetic variations. On the other hand, the outputs for the three geomagnetic signals associated to EQs indicate many Medium and High variations as shown in Figure. 10(b-d); therefore, if these results are obtained in future data they could be associated to seismic activities.

**Figure 10.** FL-based diagnosis for the analysed cases: (a) calm period and (b-d) seismic activities associated to ULF geo‐ magnetic variations.

#### **5. Conclusions**

**4.3. Fuzzy logic-based system**

120 Wavelet Transform and Some of Its Real-World Applications

according to a particular objective.

magnitude is NF (dark gray), and so on.

**Geomagnetic Fluctuations Magnitude**

**Table 2.** Rules table for the proposed FL

**Variance (Mx)**

Once the variance of the EWT signals is computed, a FL-based system is used for automatically diagnosing the severity of the ULF geomagnetic variations associated to seismic events. A FL system represents a group of rules for reasoning under uncertainty in an imprecise or fuzzy manner. It is usually used when a mathematical model of a process does not exist or does exist but is too difficult to encode and too complex to be evaluated fast enough for real time operation. Besides, it can use several sources of information in order to take a decision

The designed and implemented FL system to perform the diagnosis process is a Mamdanitype fuzzy inference system with two inputs, one output, and 16 rules. The system uses Max– Min composition, and the centroid of area method for defuzzification. The inputs are the variance of EWT results for the signals Mx and My, the Mz signal is not considered since it presents a low difference between seismic activity and calm period. These inputs are parti‐ tioned into four trapezoidal membership function sets, as shown in Figures. 9(a) and 9(b). They are labelled as NV (normal variance), LV (low variance), MV (medium variance), and HV (high variance). The output is also divided into four trapezoidal membership functions as shown in Figure. 9(c), their labels are NF (normal fluctuations), LF (low fluctuations), MF (medium fluctuations), and HF (high fluctuations). The crisp output of the Mamdani FL system can assume values between 0.5 and 4.5, where normal variations = 1, low variations = 2, medium variations = 3, and high variations = 4. The parameters of membership functions are determined according to the interpretation of the variance results by the authors. The set of rules that classifies the inputs variance is show in Table 2; there, one rule can be read as follows if (variance Mx is NV and variance My is NV) (light gray) then the geomagnetic fluctuations

**Variance (My)**

NV LV MV HV

NV NF NF LF MF LV LF LF MF HF MV LF MF HF HF HV MF HF HF HF

**Figure 9.** Membership functions: (a) Input Mx variance, (b) Input My variance, and (c) Output diagnosis.

In this chapter, a new time-frequency study based on the EWT for analysing ULF geomagnetic signals, at Juriquilla station in Queretaro Mexico, is presented. In order to prove the effective‐ ness of the EWT algorithm; three real data of EQs are analysed. The results demonstrate that the proposal has a greater detectability than DWT for detecting anomalies before, during, and after the main shock since EWT allows selecting narrow time-frequency sub-bands, unlike the DWT where the calculated time-frequency sub-bands depend on sampling frequency of the time signal. Further, the variance, a statistical complementary analysis, shows that a seismic event can be detected from 8 to 5 hrs before it occurs. It also indicates that relevant information can be obtained from 563 to 1473 km (epicenter distance) to the testing station. Therefore, the proposed time-frequency analysis can extract the abnormal signals in the ULF range of the EP related to different stages of the EQ preparation. Finally, the proposed FL system can auto‐ matically classify the magnitude variations of the EP into Normal, Low, Medium, and High variations using both Mx and My signals, where is found that Medium and High variations could be associated to seismic activities.

In a future work, the overall methodology will be implemented into a digital signal processor (DSP) for online and continuous monitoring of ULF geomagnetic variations and possibly used for obtain a implicit correlation between the seismic magnitude and the ULF geomagnetic signals.

### **Author details**

Omar Chavez Alegria1 , Martin Valtierra-Rodriguez2 , Juan P. Amezquita-Sanchez1\*, Jesus Roberto Millan-Almaraz2 , Luis Mario Rodriguez3 , Alejandro Mungaray Moctezuma3 , Aurelio Dominguez-Gonzalez1 and Jose Antonio Cruz-Abeyro4

\*Address all correspondence to: jamezquita@uaq.mx

1 Engineering Department, Autonomous University of Queretaro, Queretaro, Mexico

2 Physics and Mathematics Department, Autonomous University of Sinaloa, Cualiacan, Sinaloa, Mexico

3 Engineering Department, Autonomous University of Baja California, Baja California, México

4 Geoscience Centre, National Autonomous University of Mexico, Juriquilla, Queretaro, Mexico

#### **References**


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**Author details**

Sinaloa, Mexico

Mexico

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World Journal 201420141-8

nal2014; 20141-8.

Omar Chavez Alegria1

Jesus Roberto Millan-Almaraz2

122 Wavelet Transform and Some of Its Real-World Applications

Aurelio Dominguez-Gonzalez1

\*Address all correspondence to: jamezquita@uaq.mx

, Martin Valtierra-Rodriguez2

, Luis Mario Rodriguez3

1 Engineering Department, Autonomous University of Queretaro, Queretaro, Mexico

2 Physics and Mathematics Department, Autonomous University of Sinaloa, Cualiacan,

3 Engineering Department, Autonomous University of Baja California, Baja California, México

4 Geoscience Centre, National Autonomous University of Mexico, Juriquilla, Queretaro,

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## *Edited by Dumitru Baleanu*

The book contains six chapters. The use of the progressive regressive strategy for biometrical authentication through the use of human gait and face images was investigated. A new lossy image compression technique that uses singular value decomposition and wavelet difference reduction technique was proposed. The best wavelet packet based selection algorithm and its application in image denoising was discussed. The scaling factor threshold estimator in different color models using a discrete wavelet transform for steganographic algorithms was presented. The extraction of features appearing in current signal using wavelet analysis when there is rotor fault of eccentricity and broken rotor bar was debated. The application of the empirical wavelet transform for seismic anomalies detection in ultralow-frequency geomagnetic signals was illustrated.

Photo by Egor Lisovskiy / DollarPhoto

Wavelet Transform and Some of Its Real-World Applications

Wavelet Transform and Some

of Its Real-World Applications

*Edited by Dumitru Baleanu*