1. Introduction

2. Data and Preprocessing

3. Comprehensive Methodological Approach

3.1. Wavelet Analysis Method

The fundamental principle of wavelet analysis is signal decomposition through a series of wavelet functions that adapt to different frequency ranges and time frames. By decomposing the signal into multiple wavelet functions, useful spatial characteristics can be extracted, and non-linear and non-stationary characteristics can be accurately and efficiently drawn out. Meanwhile, wavelet analysis can also reconstruct the time-domain and frequency-domain characteristics, therefore obtaining more complex information regarding signal quality.

Specifically, wavelet analysis considers seismic signals as the overlap of a set of wavelet functions, describing local seismic signal traits at different scales. By analyzing these wavelet function coefficients, seismic signal characteristics such as amplitude, frequency, phase can be extracted, which can be used to predict the occurrence of earthquakes.

3.1.1. Continuous Wavelet Base Function

The choice of the wavelet base function determines the efficiency and effect of wavelet transformation. The wavelet base function is flexible to choose and can be constructed based on the issues faced. The following list includes a few continuous wavelet base functions employed in this paper:

(1)

Haar Wavelet

(1)

It can be said that the Haar wavelet is the simplest of all known wavelets, as shown in the figure. For the translation of t, the Haar wavelet is orthogonal. For one-dimensional Haar wavelet, it can be regarded as the completion of the difference operation, that is, the difference of the part which is not equal to the average value of the observed results. Obviously, Haar wavelets are not continuous differentiable functions.

(2)

Mexico Wavelet

The Mexico hat wavelet is the second derivative of the Gaussian function, that is:

(2)

The coefficient $\frac{2}{\sqrt{3}}{\pi }^{-1/4}$ is mainly to ensure the normalisation of ψ(t), i.e., ${\Vert \psi \Vert }^2=1$. This wavelet uses the second order derivative of the Gaussian smoothing function and is named due to the similarity of the waveform to the Mexican Hat(straw hat) parabolic contour line. It has gained more applications in visual information processing research and edge detection, hence it is also called as Marr wavelet.

A cluster of wavelets can be given by the $m$th order derivative of the Gaussian function:

(3)

The constant in Equation (3) is to ensure that ${{\psi }_m}^2=1$. Although the Mexican straw hat wavelet is equivalent to the case of m=2 in Equation (3), it is isotropic and thus cannot detect different directions of the signal.

The DOG wavelet formed with the Difference of Gaussians (DOG) function is a good approximation of the Mexico straw hat wavelet as

(4)

(3)

Morlet real wavelet

(5)

(4)

Morlet complex-valued wavelet

The Morlet wavelet is the most commonly used complex-valued wavelet, which is defined by Equation (6).

(6)

The Fourier transform of Equation (6) is given by

(7)

Typically, ${\omega }_0\ge 5，{\omega }_0=5$ is the most used case. $f_B$ is the bandwidth and $f_C$ is the centre frequency.

(5)

Complex Gaussian wavelet

A complex Gaussian wavelet consists of the nth order derivative of a complex Gaussian function defined as follows:

(8)

The constant $C_n$ is used to maintain the energy normalisation property of the wavelet function.

(6)

Fushanong wavelet

(9)

where $f_B$ is the bandwidth, $f_C$ is the centre frequency and m is a positive integer.

3.1.2. Wavelet Maxima

Let $W_f\left(s,x\right)$ be the convolutional wavelet transform of the function $f\left(x\right)$, and the point $\left(s_0,x_0\right)$ is said to be a local extremum point under the scale $s_0$ if $\frac{\partial W_f\left(s_0,x\right)}{\partial x}$ has a point over zero on $x_0$. If $\left(s_0,x_0\right)$ has a zero crossing point on, then is the modulo maximal point of the wavelet transform, and for any point x belonging to a neighbourhood of x₀, there are

(10)

Mathematically, an infinitely derivable function is said to be smooth or free of singularities, and a function is said to be singular here if it has a break somewhere or if the derivatives of a certain order are discontinuous. A mutating signal must be singular at its point of mutation.

The degree of singularity of a function (or signal) $f\left(x\right)\in R$ at a point is often portrayed by its singularity index, Lipschitz $\alpha$, abbreviated as Lip index.

Let $0\le \alpha \ge 1$, at the point $x_0$ if there exists a constant K, for any point $x$ in the neighbourhood of $x_0$ such that the following equation holds:

(11)

Then $f\left(x\right)$ is said to be Lip$\alpha$ at $x_0$.

The Lip index $\alpha$ gives precise information about the conductivity of the signal $f\left(x\right)$ at the point $x_0$. If $\alpha =1$, the function $f\left(x\right)$ is continuously differentiable at $x_0$, the function $f\left(x\right)$ is said to have no singularity; If $\alpha =0$, the function $f\left(x\right)$ is interrupted at $x_0$; If $0<\alpha <1$, the smoothness of the function decreases. α larger, that the singular function $f\left(x\right)$ closer to the rule; α smaller, that the singular function $f\left(x\right)$ at $x_0$ point changes the more sharp. Usually, the singularity of a signal tends to behave as a positive singularity, while noise behaves as a negative singularity.

The use of wavelet transforms to detect the singularity of the signal is mainly reflected in two aspects: one is the use of wavelet transform on the location of the singularity of the positioning and noise cancellation processing; the second is the use of wavelet transform to determine the singularity of the singularity index (i.e., the size of the singularity).

The wavelet used for singularity detection is different from the general orthogonal wavelet, which is obtained from the smooth function. Let $\theta \left(x\right)$ be a smooth function with low-pass property, and take its first-derivative ${\phi }^{\prime }\left(x\right)=\frac{d\theta \left(x\right)}{dx}$ as a wavelet, it is not difficult to prove the following relationship:

(12)

where: ${\theta }_s\left(x\right)=\frac{1}{s}\theta \left(\frac{x}{s}\right)$. The above equation shows that the 3-wavelet transform of the signal $f\left(x\right)$ can be expressed as the first-order derivative of the signal after $f\left(x\right)$ is smoothed by ${\theta }_s\left(x\right)$ at the scale s, where the extreme point of the wavelet transform is the turning point of $f\left(x\right)·{\theta }_s\left(x\right)$, which in the limit is the step point. The advantage of characterising the signal in terms of the extremes of $W_f^1\left(s,x\right)$ is that it is possible to distinguish between the very large and the very small. Extremes are where the signal changes dramatically and minima are where the signal changes slowly. These basic conclusions form the theoretical basis for the detection of anomalies (sudden signal change points) by the wavelet extreme value method.

4. Earthquake Case Studies

5. Results and Discussion

6. Conclusions

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