Ground Penetration Radar is a controlled source geophysical method which uses high frequency electromagnetic waves to study shallow layers. Resolution of this method depends on difference of electrical properties between target and surrounding electrical medium, target geometry and used bandwidth. The wavelet transform is used extensively in signal analysis and noise attenuation. In addition, wavelet domain allows local precise descriptions of signal behavior. The Fourier coefficient represents a component for all time and therefore local events must be described by the phase characteristic which can be abolished or strengthened over a large period of time. Finally basis of Auto Regression (AR) is the fitting of an appropriate model on data, which in practice results in more information from data process. Estimation of the parameters of the regression model (AR) is very important. In order to obtain a higher-resolution spectral estimation than other models, recursive operator is a suitable tool. Generally, it is much easier to work with an Auto Regression model. Results shows that the TQWT in soft thresholding mode can attenuate random noise far better than TQWT in hard thresholding mode and Autoregressive-FX method.
Ground Penetration Radar is a controlled source geophysical method which uses high frequency electromagnetic waves to study shallow layers. The use of this method began in 1956 and has been developed since the 1970s. GPR system has been commercially available since the 1980s, and has been widely used since the mid-decade. In the GPR method, high frequency electromagnetic waves (from 12.5 to 2300 MHz) transmit to the earth and these waves are reflected by objects or clear boundaries between the underground layers. Electromagnetic reflections are created by the difference in electrical conductivity (dielectric constant) between materials that pass electromagnetic waves. GPR electromagnetic waves pass through materials that are electrically conductive, but they are heavily absorbed when passing through high-conductivity materials such as clays, organic acid soils and saturated saline water [
Resolution of GPR varies from depth of centimeters to a several meters with maximum depth of about 100 meters. Resolution of this method depends on difference of electrical properties between target and surrounding electrical medium, target geometry and used bandwidth. Resolution of this method can be high which can recognized fine layers in near surface structures and objects in the ground can be well [
The wavelet transform is used extensively in signal analysis and noise attenuation. In addition, wavelet domain allows local precise descriptions of signal behavior. The Fourier coefficient represents a component for all time and therefore local events must be described by the phase characteristic which can be abolished or strengthened over a large period of time. Wavelet expansion coefficients represents a component which is local itself and convenient for interpretation. The wavelet Transform may allow separation of signal components which overlaps in time and frequency. Wavelets can be designed to fit the unique applications because they are also customizable and tunable and there is not just one wavelet. They are ideal for adaptive devices that adjust themselves to the signal. Finally, production of wavelets and the calculation of discrete wavelet transforms (DWT) are well suited to digital computing [
Finally basis of Auto Regression (AR) is the fitting of an appropriate model on data, which in practice results in more information from data process. Estimation of the parameters of the regression model (AR) is very important. Also use of a suitable model is very important in approximation of random signals spectrum, especially in parametric methods. Parametric methods of power spectrum estimation are based on selecting the appropriate model for the data [
In order to obtain a higher-resolution spectral estimation than other models, recursive operator is a suitable tool. Generally, it is much easier to work with an Auto Regression model. For this reason, suitable techniques are presented for estimating of model parameters [
GPR signal in receiver X(t) can be formulated as equation (usually infected with noises):
X ( t ) = S ( t ) ∗ ω ( y ) + n ( t ) (1)
In Equation (1), the earth reflection series s(t) is matched in the wavelet due to the spring w(t) and the noise n(t) is added to the data, which must be corrected. Since it is impossible to eliminate all of the noise, then the denoising is done with the aim of obtaining X(t) is as close as possible to s(t) [
In this research, noise attenuation method of auto regressive on wavelet and f-x domains was used to reduce the noise of the GPR signal. First AR operator and definition of f-x domain will illustrate and explained using some examples. Ultimately, all methods will be applied to real and synthetic data of the GPR signal and their results will be compared together.
In this section, a wavelet transform method is described where it easily uses the Q factor and wavelet transform process can be controlled using the periodic nature of the signal. The function of scale has real values. The Q factor is a ratio of the frequency center to signal bandwidth. Most wavelet transform methods have a poor ability to control behavior of signal frequencies. The TQWT method is completely discrete, it has a complete signal reconstruction ability and mean redundancy.
The low-frequency region of the signal is analyzed using the low-pass function. If the scale parameter α is chosen for the low-pass scale function part, the sampling changes to α f s , where f s is sampling frequency of the input signal. If 0 < α ≤ 1 , then the function of the low-pass signal changes:
Y ( ω ) = X ( α ω ) , | ω | ≤ π (2)
And if, low pass function changes a signal like:
Y ( ω ) = { X ( α ω ) , | ω | ≤ π / α 0 , π / α < | ω | < π (3)
Note that low-pass function keeps signal behavior around a zero frequency according to
The high-frequency signal region is analyzed by high-frequency scale function. If the scale parameter β is selected for the high-pass section function, the sampling is changed to β f s , which f s is sampling frequency of input signal. If 0 < β ≤ 1 , the of the high-pass scale function will change such a signal:
Y ( ω ) = { X ( β ω + ( 1 − β ) π ) , 0 < ω < π X ( β ω − ( 1 − β ) π ) , − π < ω < π (4)
And if 1 ≤ β , high pass scale function changes like:
Y ( ω ) = { 0 , | ω | < ( 1 − 1 / β ) π X ( β ω + ( 1 − β ) π ) , ( 1 − 1 / β ) π < ω < π X ( β ω − ( 1 − β ) π ) , − π < ω < ( 1 − 1 / β ) π (5)
The high-pass scale function keeps signal around the Nyquist frequency according to
Using a frequent dual channel filter, the TQWT method is applied. This transform is represented by three steps in
Using a frequent dual channel filter, the TQWT method is applied. This transform is represented by three steps in
ω ( j ) ( n ) denotes the subband in jth step and it’s generated for j = 1 high-pass subband in the first stage. The subband bandwidth of the jth frequency is α j − 1 f s . Also f s is a sampling frequency of input data.
The dual-channel filter in
r = β 1 − α (6)
This relation is deduced from the fact that a sampling frequency under the jth subband for j ≥ 1 is determined by β α j − 1 f s , So sampling frequency is equivalent to (6).
The frequency response in the jth step H 1 j ( ω ) is nonzero at the distance ( ω 1 , ω 2 ) .
ω 1 = ( 1 − β ) α j − 1 π , ω 2 = α j − 1 π , (7)
The central frequency of the jth stage is approximately average ( ω 1 , ω 2 ) .
ω 0 = 1 2 ( ω 1 + ω 2 ) = α j 2 − β 2 α π (8)
This relationship is based on the sampling frequency:
f c = α j 2 − β 4 α f s (9)
The bandwidth of the frequency response under the jth subband is about a half of bandwidths of all frequencies that have a non-zero response.
B W = 1 2 ( ω 1 − ω 2 ) = 1 2 β α j − 1 π (10)
If the Q factor is expressed on the basis of α , β
Q : = ω c B W = 2 − β β (11)
Note that Q is not dependent on the stage. A Transform method is Q constant and depends only on filter bank parameters. To determine α and β, they are rewritten based on r and Q:
β = 2 Q + 1 , α = 1 − β r (12)
A Q factor must be greater or equal to 1, and if Q is equal to 1, wavelet transform builds a second derivative of a Gaussian function. If Q is greater than one that means signal has a higher periodic behavior and the Oversampling rate should be greater than one. If r is near 1, the transition region is very thin and the time response of the wavelet is not properly determined. If r ≈ 1 makes a sinc wavelet, then it’s therefore optimal to r ≥ 3 .
When the signal is staircase oscillatory, a number of vanishing moments of the wavelet transform is considered. This is not the case when we talk about a periodic signal like a GPR signal. The number of steps for this type of wavelet transform is limited by the length of the signal. After a certain stage, the signal will be very short to become two samples. If countless steps are taken, conversion of these steps is difficult to reconstruct. J must be determined so that the wavelet does not extend beyond the length of the category.
J max = ⌊ log ( β N 8 ) log ( 1 α ) ⌋ (13)
Generally, a signal in time domain is seen as a mixed signal in frequency domain. From frequency domain, information such as phase spectrum, the amplitude spectrum and energy spectrum can be obtained. The energy distribution is also extracted in terms of frequency [
E = 1 2π ∫ + ∞ − ∞ | x ( f ) | 2 d f (14)
Given that in time domain, input data or signal, for each frequency is shown as sinusoidal functions in the location, the separation of the signal from the noise is made easier. Therefore, we will investigate predictive filter in this domain. This method assumes that a class is composed of delayed impacts in accordance with
The f-x prediction filter method is very useful method to attenuate random noise. 1. Selesnick, (I.W. Sparse signal representations using the tunable Q-factor wavelet transform. in Wavelets and Sparsity XIV. 2011. International Society for Optics and Photonics.)
In this method, it is assumed that trends are linear. The seismic section can be divided into smaller windows in cases where trends are not linear, in order to satisfy the assumption given that trends in that window are linear. Assume that a seismic U ( x , t ) is a sequence of impacts with different domains defined as [
U ( x , t ) = ∑ j = 1 N A j δ ( t − g j ( x ) ) (15)
where t is the time, x is the lateral position, A i is the jth impact, and g j ( x ) is the delay function that represents the event’s form. Considering Fourier transform of (15) we have:
U ( x , ω ) = ∑ j = 1 N A j e − i ω g j ( x ) (16)
Since exponential functions can be written as a set of sinuses and cosines, in (16), the hypothesized seismic traces are converted into a set of sinuses and cosines as functions of ω and x. Since the f-x filter only predicts linear data, events should be linear to x. So the function g j ( x ) must be linear [
U ( x , ω ) = ∑ j = 1 N C j e − i ω b j x (17)
C j is a conjugate constant, which depends on the source power and the reflection coefficient, and B j is the gradient of the linear event. (17) Shows that by linearization assuming, the function U ( x , ω ) is completely sinusoidally in x, which means that this function is predictable and predictable. Therefore, the signal is a complex exponential function in terms of x in the time-frequency domain and it will be predictable [
Classical models of time series are divided into stationary and non-stationary sections. Autoregressive vector is part of classic static time series model that we will discuss in this article. One of the limitations of our models is that they impose a one-way relationship whose predictor variables are influenced by predicted variables which this process must be reversed. However, in many cases, the reversal of this action must also be made where the variables affect each other. In this framework, all variables behave symmetrically [
y t = c + a 1 y t − 1 + ⋯ + a p y t − p + ε t (18)
where c is constant and ε t is a white noise with mean zero and the variables σ ε 2 and a i are the parameters of the model. In this case, y t is called the p-th Autoregressive model with and is represented by AR (p). An Autoregressive structure is simple, useful, and easy to understand in a wide range of fields. The first-order Autoregressive is defined as follows [
y t = a 1 y t − 1 + ε t (19)
Assumptions:
Firstly, the residue is expected to be zero
E [ ε i , t ] = 0 with i = 1 , 2 (20)
And secondly, errors are not self-correlated.
E [ ε i , t ε j , τ ] = 0 with t ≠ τ (21)
Automatic regression models are considerably flexible in controlling a wide range of different time series.
Autoregressive models do not allow us to explain on causal relationships. This is especially applies when Autoregressive models are merely generalized for the processing of unknown time series. While a causal interpretation requires a basic cost model. However Autoregression provides a proper interpretation of the variables shown [
In order to increase the signal-to-noise ratio of a multi-point signal (S/N), firstly the Autoregression vector of the noisy data is estimated and shown as a A ^ j . Then, the forward estimated of denoised data is obtained by the following equation [
g ^ k f = ∑ j = 1 M A ^ j g k − j , k = M + 1 , ⋯ , N (22)
Similarly, the estimated inverse equation of de-noised data are obtained:
g ^ k b = ∑ j = 1 M A ^ j * g k + j , k = M + 1 , ⋯ , N − M (23)
A ^ j * represents the complex conjugate of Autoregressive vector operator. The final equation of the de-noised data are obtained by mean for forward and inverse estimates:
g ^ k t = g ^ k f + g ^ k b 2 (24)
Given that the purpose here is to reduce the random noise, in the first step a random noise should be added to the section which obtained from synthetic models. For synthetic data generation, the ground model is firstly considered with arbitrary coefficients. Given the similarity in wave propagation of electromagnetic waves and seismic waves, synthetic modeling of GPR and seismic data are analogous. Random noise is added to synthetic model and various de-noising methods which presented in this article apply to it. After denoising of synthetic model, real GPR data will be denoised. Here, by applying linear Autoregression modeling on wavelet and f-x domain, random noise in actual data will be weakened. The procedure is applying these steps on real data in wavelet domain and then transforms the denoised data waveform from wavelet domain. After these steps and in next section, results will be compared and evaluated.
The wavelet used to generate this synthetic data is the Ricker Wavelet. This wavelet is formulated as [
ω ( t ) = ( 1 − π 2 f 2 t 2 ) exp ( − π 2 f 2 t 2 ) (25)
ω ( t ) is Ricker wavelet, t is time and f is the central frequency. Ricker wavelet is symmetric in time domain and has zero mean ( ∫ − ∞ + ∞ r ( τ ) d τ = 0 ).
The generated synthetic signal is formulated as:
X ( t ) = S ( t ) ∗ ω ( t ) + n ( t ) (26)
In (26), S ( t ) is a received that is correlated with the Ricker wavelet (as a synthetic wavelet source). n ( t ) is a random noise that is added to the input signal.
To evaluate efficiency of mentioned denoising methods, an artificial section considered which is consisting of five layers and three pipes according to
The only difference between noisy and the hard thresholding denoised signals are in a deformation of the noises. The noises are somewhat diminished, but it shows another form of noise as an elongation along the axis of time. The boundaries of layers and hyperbolas are still confusing and effect of noise is not diminished.
High-frequency noises have been far more attenuated, but the elongation portion is also included in this filter. But this elongation is significantly less than the hard thresholding. Therefore, signal is significantly improved by soft thresholding method compared to the hard thresholding.
To denoising in this section, Autoregressive filtering is applied in time domain.
In this section the filtering on real data will be discussed. Firstly the Autoregressive-FX filter and after that the TQWT in soft and hard thresholding ways.
With respect to
As stated several factors make the GPR data noisy such as mobile networks, power stations, etc. Considering that the increase of signal-to-noise ratio is the main objective of this research a method was chosen that to attenuating the noise and minimizes the damage to the signal. In this paper synthetic and real GPR data was denoised with the TQWT in two soft and hard thresholding and the Autoregressive-FX filter. According to the results, it can be concluded that the hard thresholding method is not suitable. Results do not show a good improvement, especially at low frequencies, most of the noise remains, and the elongations which caused by hard threshold has also been added to these noises.
However, in the soft thresholding problem of elongation noise is observed. But a significant part of the noise is attenuated and high-frequency noise has been eliminated. After the deactivation, the spikes from the real signal are well observed. Also boundaries of layers and hyperbolas are also returned. The Autoregressive-FX method has succeeded to denoising of high-frequency noise. The layer boundaries are also clear, but less denoising has been achieved compared to the soft thresholding TQWT. Also layer boundaries are less detected and part of high-frequency noise is still seen.
As a result, the wavelet domain provides a suitable frequency resolution for noise attenuation. But the type of filter applied to the wavelet transform also affects denoising. The TQWT method has a good frequency resolution because it operates in the wavelet domain, but the results of hard thresholding indicate that the selection of the appropriate filter for de-noising substantially changes results. Finally, by comparing all of results, it can be said that the TQWT soft thresholding method has been much more successful compared to the hard thresholding and Autoregressive-FX in denoising and converting the actual signal.
The authors declare no conflicts of interest regarding the publication of this paper.
Bardar, A.E., Oskooi, B. and Goudarzi, A. (2019) Comparison of GPR Random Noise Attenuation Using Autoregressive-FX Method and Tunable Quality Factor Wavelet Transform TQWT with Soft and Hard Thresholding. Journal of Signal and Information Processing, 10, 19-35. https://doi.org/10.4236/jsip.2019.101003