Stratigraphic Absorption Compensation of GPR Signal Based on Improved S-transform* Kunwei Feng, Yonghui Zhao*, Zaiyuan Zhang
Shuangcheng Ge
School of Ocean & Earth science Tongji University Shanghai, China
[email protected]
Zhejiang Institute of Hydraulics and Estuary Hangzhou, China
Abstract—The loss of the higher frequency components of electromagnetic wave due to soil absorption and attenuation results in GPR signal decrease rapidly in deep areas. High frequency compensation is one of most important ways to improve GPR data resolution. The S-Transform (or ST) has been widely used in digital signal processing since it was proposed. Here, improved S-transform to GPR data has been designed to enhance the deep reflection events. Firstly, GPR signals are transformed to S-transform coefficients trace by trace; then at each time sampling point, the frequency factors depended attenuation are extracted according to the soil layer feature; the S-transform coefficients are weighed by the factors to ensure energy equalization at different time sampling point. Numerical simulated and real GPR data are used for testing the proposal and its validity. According to the time-frequency spectrum analysis before and after ST treatment, almost the same energy distribution in lower frequency range is obtained, and the higher frequency components are obviously increased at deep area. The comparison demonstrates that the reconstructed GPR data from S-transform with appropriate weighting method can implement compensation absorption very well without Q value. Keywords—Ground penetrating radar; compensation; S-transform; time-frequency analysis
I.
frequency
INTRODUCTION
GPR is a kind of nondestructive testing technologies, which aims at the localization of invisible targets or interfaces in objects or underground. It takes incomparable advantage over traditional geophysics methods and is widely used in detecting shallow depth objects. But meanwhile with burial depth increasing, the target reflection interface cannot be distinguished in waveform figure. The detecting error increases with the object depth and the reflection wave strength will greatly decrease. In engineering practice, lower center frequency means a great detection range but the resolution decreased at the same time. The conflict between resolution and detection range has been the main barrier for a successful GPR application. It should be a reasonable solution to increase the intensity of reflection wave of deep interfaces from the GPR profile collected using higher center frequency antenna by applying some methods. Generally speaking, automatic gain control (AGC) can be used to enhance GPR weak signal of deep target. But when the dielectric coefficient and electrical conductivity of underground media varies significantly, neither the AGC nor the SEG method seem
likely difficult to obtain better results. The S-transform has gained a major interest in time-frequency representation due to its special features over the conventional wavelet transform [1]. Stockwell proposed the S-transform, which is a nonstationary signal analysis and processing method [2]. Comparing with other techniques of the time-frequency analysis, for instance, the Short Time Fourier (SFT), Gabor Transform (GT) and the Continuous Wavelet Transform (CWT) [3,4,5], the S-transform has its unique advantages. Firstly it provides not only a frequency dependent resolution but also has a direct relation to Fourier Transform. Secondly the S-transform is theoretically perfectly invertible [2]. Besides wave reflection, the amplitude of the electromagnetic wave includes the effect of wave front diffusion, stratigraphic absorption, attenuation, and transmission energy loss, etc. With increasing of the frequency and penetration depth, absorption and attenuation dominate and lead to energy attenuation. The stratigraphic absorption compensation is an effective method to compensate for the absorption of the high frequency components, making the amplitude spectrum of the radar record similar to that of reflectivity in the frequency range [6]. Generally, the inverse Q filtering is mostly used to compensate for stratigraphic absorption, but it is necessary to obtain the precise Q values. Actually the calculated Q values are not accurate, which limits the application of the method [7]. In this paper, based on the characteristics of radar wave variation with time, space and frequency, high resolution processing of electromagnetic data with the S-transform has been applied in the GPR amplitude spectrum compensation. Numerical simulation and real GPR data application shows that the S-transform method proved to be an effective method to compensate for the stratigraphic absorption. II.
BASIC PRINCIPLES
A. The S-Transform The S-transform of h(t) is defined as: +∞
S (τ , f ) = ∫ h(t ) −∞
|f| ⎡ 1 ⎤ exp ⎢− (τ − t ) 2 f 2 ⎥ exp(−2πift )dt 2 2π ⎣ ⎦
(1)
where τ and f correspond to time and frequency. The basic wavelet function is
The research is sponsored by Natural Science Foundation of China (No. 41374146) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
978-1-4799-6495-6/15/$31.00 ©2015 IEEE
⎛ t2 f 2 ⎞ |f | − i 2πft ⎟⎟ Wf ( f ) = exp⎜⎜ − 2 2π ⎝ ⎠ = g r (t ) exp(− 2πift )
xˆ ( f ) = rˆ( f , Tk )ωˆ ( f )
| X ( f , Tk ) |=| ωˆ ( f )rk Ak ( f , Tk ) | (3)
the Fourier transform h(f) of h(t) is obtained by calculating the integral along the time shift variable τ −∞
S (τ , t )dt = H ( f )
(4)
+∞ +∞
∫
−∞ −∞
S (τ , f )dτ exp(2πift )df
ϕ ( f , Tk ) = 2πfTk
B. Absorption compensation based on the S-Transform In practice, during the electromagnetic wave propagation, owing to absorption, the wavelet amplitude gradually attenuates and the received GPR data include this attenuation. In fact, the soil medium is a low-pass filter which varies with time, namely the Q filter, including the convolution effect. Supposed the stratigraphic reflection coefficient is represented by rk (k=0,1,...,K), the amplitude response after Q filter at each Tk corresponding to each rk is as follows,
Ak ( f , Tk ) = A0 ( f ,0) exp[−2πfTk / Qeq (Tk )]
(6)
where Qeq(Tk) is equivalent Q value at Tk, A0(f, 0) is equivalent amplitude at initial time. Assuming only the stratigraphic absorption results in amplitude attenuation, the reflectivity function can be written as follows
⎧r (T ) t = Tk r (t )δ (t − Tk ) = ⎨ k else ⎩ 0
(7)
the Fourier transform is
∫
+∞
−∞
r (t )δ (t − Tk ) exp(−2πift )dt K
= ∑ rk exp(−2πifTk )
(8)
Considering amplitude attenuation, the Fourier transform of reflectivity function is expressed as K
Supposed radar wavelet is ω(t), and the Fourier transform is ωˆ ( f ) , the radar reflect record can be written as
The form in frequency domain is
then divided by the attenuation ratio α(f0,Tk) at both sides of the equation
α w ( f , Tk ) =
| α ( f , Tk ) | exp(−2πfTk / Qeq (Tk ) = | α ( f 0 , Tk ) | exp(−2πf 0Tk / Qeq (Tk )
(15)
Use the 1/αw(f, Tk) weighting X(f, Tk) to eliminate the stratigraphic absorption attenuation, then reconstruct radar time section by inverse S-Transform. C. Calculation flow To the 2D radar data h(nt,nx), the S-transform process steps is, 1) Do the S-transform to a each trace x(nt) radar data, to obtain the time-frequency distribution |X(f, Tk)|, where nt is number of the single trace data, k=0,1,2,...,nt-1; 2) Use (14) to calculate α(f, Tk) at each frequency component k=0,1,2,...,K; 3) Use (15) to obtain the weighting ratio αw(f, Tk) k=0,1,2,...,K; 4) Take the reciprocal of αw(f, Tk) weight the timefrequency distribution for the corresponding frequency at time sampling point t, that is X(f, Tk), k=0,1,2,...,K;
6) Then repeat the processes from step 1) to 5) to all the GPR data trace by trace, to get GPR profile after absorption compensation.
(9)
k =0
x(t ) = r (t )δ (t − Tk ) * ω (t )
| X ( f , Tk ) | | ωˆ ( f )rk Ak ( f , Tk ) | = (14) | X ( f ,0) | | ωˆ ( f ) A0 ( f ,0) |
5) Use the results of the compensation at each frequency to reconstruct the radar record in the time domain.
k =0
rˆ( f , Tk ) = ∑ rk Ak ( f , Tk ) exp(−2πifTk )
(13)
According to (1), the ratio of reflection wave amplitude at each frequency to the initial time is
α ( f , Tk ) = (5)
(12)
the phase spectrum is
thus, the inverse S-transform is
h(t ) = ∫
(11)
where the amplitude spectrum of reflection wave at Tk is
⎛ t2 f 2 ⎞ |f| ⎟ exp⎜⎜ − g (t ) = 2 ⎟⎠ 2π ⎝
∫
= ωˆ ( f )∑ rk Ak ( f , Tk ) exp(−2πifTk ) k =0
the Gaussian window is
+∞
K
(2)
(10)
III. A.
EXPERIMENTS AND ANALYSIS
Synthetic GPR data To test the proposed S-transform algorithm, synthetic GPR records were generated using GPRMAX2D, a FDTD simulation method in 2D space [8]. Based on a high stratigraphic absorption case, a four-layers structure was designed to simulate a concrete layer, a riprap layer, transition
TABLE I.
DIELECTRIC AND GEOMETRIC PARAMETERS OF MEDIA
Conductivity
(m)
Relative dielectric constant
concrete
0.5
9
0.005
Medium
Thickness
(S·m-1)
clay
3.3
25
0.1
transition layer
0.9
12-25
0.005-0.1
rock
-
9.7
0.015
compensation, particularly by highlighting the signal events in deep region. After the absorption compensation, the resolution is improved and the energy is recovered. Shallow reflections had barely changed, and the deep events invisible on the original profile are shown after the absorption compensation. Fig. 4a and Fig. 4b are the single trace random selected from GPR record before and after processed. Comparison between these two figures demonstrates that the amplitude has been strengthened after 200 ns. Fig. 4c and Fig. 4d shows the time frequency spectrum of the 20th trace before and after the compensation. After the compensation, GPR signal energy at the deep layer (range from 200 ns to 480 ns) had been greatly improved, and the dominant frequency band has been widened. 4 Amplitude
layer and clay layer. Four circles of different dimensions were included to represent isolated rocks in clay layer (Fig. 1). The dielectric and geometric parameters of the media used in the model are listed in Table 1. The simulated 2D GPR profiles (Fig. 2) show noticeable reflections from the upper rocks, but reflections from deep interfaces and rocks are very weak. Stransform has been applied to compensation deep signals to highlight the deep reflections in the GPR time section.
2 0 -2
0
100
200
300
400
500
300
400
500
Time (ns) (a)
Amplitude
200 100 0 -100
0
100
200
Frequency (MHz)
Fig. 1. Numerical model of costal embankment
Frequency (MHz)
Time (ns) (b)
Fig. 2. Synthethc GPR image using 50MHz (15ns AGC time window)
200
1 0.8
100
0.6 0.4
0
0
100
200 300 Time (ns) (c)
400
500
200
0.2
20 15
100
10 0
5 0
100
200 300 Time (ns) (d)
400
500
Fig. 4. The contrast of single traces and time-frequency spectrums (trace 20th). (a) Single trace of gained data, (b) single trace after compensation, (c) time frequency spectrum of gained data, (d) time frequency spectrum after compensation.
Fig. 3. GPR image after frequency compensation using improved ST
A comparison of the GPR profile after S-transform (Fig. 3) with the original simulated GPR profile (Fig. 2) reveals that Stransform is effective for the stratigraphic absorption
B. Real GPR data The proposed method is applied to a real GPR profile recorded collected by EKKO-PRO GPR system with 100 MHz antenna in reflection mode. The distance between two antennas is 1.0 m. The test site located in a construction site and it was filled with building rubbish and industrial waste. The detection aim is to find the original foundations. The test area is next to the Huangpu River and underground filled with water. Radar wave attenuates very quickly with depth in this case. Fig.5 gives the GPR profile of raw data. Because of the stratigraphic absorption, it is impossible to identify the
reflection events related to the deep interfaces or targets from the original GPR image. 4
x 10
0
3
40 2 Tracel time (ns)
80 120
1
160
0
200
-1
ns). From Fig.7, the reflection events from deep interfaces are easily identified, it might be the reflection from the original foundations top. We can easily get the two-way travel time of signals reflected from ground to the reflection interface is about 160 ns. The thickness of coverage layer d can be calculated as d=v·t/2. Considering the average velocity of radar wave in aquifer stratum is about 0.06m/ns, the thickness of coverage layer above the original foundations is approximate 4.8 m in the test site.
240 -2
IV.
280 0
5
10
15
20 25 Scan Axis (m)
30
35
40
-3
45
S-transform is extensions of CWT which based on Morlet wavelet, and has direct relation with Fourier transform. The unique advantage makes it become a powerful measure to analysis non-stationary signals. The absorption compensation method based on S-transform fully takes into account the attenuation characteristic that the absorption is sensitive to high frequency, which fits with the resolution needs of radar wave absorption compensation. According to the comparison between the time-frequency spectrum before and after process, almost the same energy distribution in lower frequency range and more wide frequency bandwidth is obtained, and the higher frequency components in GPR signal is obviously increased. The technique was effective in getting a higher resolution and more deep information from radar profile. Experiment results to simulated and real GPR data show that the method is suitable for signal processing during GPR exploration for deep buried target.
Fig. 5. Raw GPR profile 4
x 10
Tracel time (ns)
0
8
40
6
80
4 2
120
0 160
-2
200
-4
240
-6 -8
280 0
5
10
15
20 25 Scan Axis (m)
30
35
40
45
Fig. 6. GPR profile after compensation using improved ST 4
x 10 Amplitude
5 0 -5 0
50
100
150
200
250
300
a)
ACKNOWLEDGMENT
4
x 10 5 Amplitude
CONCLUSION
The authors are grateful to the reviewers for their advice for an early abstract. The authors also thank Mr. Zhaolei Jia and Mr. Liming Zhu for their support during field GPR data acquisition.
0 -5 0
50
100
150
200
250
300
b)
Frequency (MHz)
150 15000
REFERENCES
100 10000 50
0
[1]
5000
0
50
100
150
200
250
300
0
c)
Frequency (MHz)
150 15000
[2]
100 10000 50
0
5000
d) Fig. 7. The contrast between raw data and processed data at trace 200. (a) the raw data single trace, (b) after the compensation trace, (c) time-frequency spectrum of 200th trace, (d) time-frequency spectrum of 200th. 0
50
100
150
200
250
[3]
300
Fig. 6 gives the S-transform result. The shallow reflections events had barely changed after S-transform. However, the deep events can be clearly recognized from the processed GPR image. As shown in Fig.7a and Fig.7b, the comparison between the original and processed data of a random trace(the 200th trace) abstracted from GPR image, demonstrates that the reflection events from deep interface was easily distinguished. The time-frequency spectrum contrast between Fig.7c and Fig.7d reveals that the energy in shallow nearly constant, and radar energy in deep greatly increased (range from 200 to 250
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