SURFACE WAVE PREDICTION AND REMOVAL FROM SEISMIC DATA

Abstract

The present method predicts and separates dispersive surface waves from seismic data using dispersion estimation and is completely data-driven and computer automated and no human intervention is needed. The method is capable of predicting and suppressing surface waves from recorded seismic data without damaging the reflections. Nonlinear signal comparison (NLSC) is used to obtain a high resolution and accurate dispersion. Based on the dispersion, surface waves are predicted from the field recorded seismic data. The predicted surface waves are then subtracted from the original data.

Claims

1. A method for processing seismic data to remove interference from surface waves, comprising: obtaining and recording input multi-channel seismic data from a plurality of receivers, wherein the input multi-channel seismic data comprises a plurality of recorded seismic traces at a plurality of locations near the receivers; generating estimated surface wave phase velocities from the input multi-channel seismic data using nonlinear signal comparison (NLSC); generating predicted surface waves for each receiver by using the estimated surface wave phase velocities and performing a phase shift and local stacking analysis for seismic traces at locations near each receiver; and subtracting the predicted surface waves from the input multi-channel seismic data to generate seismic data lacking interference from surface waves.

2. The method of claim 1, wherein the step of generating estimated surface wave phase velocities comprises using the following equation: $S_{\mathrm{NLSC}}^{\mathrm{ij}}\left(\omega ,V_{\mathrm{ph}};\sigma \right)=\frac{S_{\mathrm{NL}}^{\mathrm{ij}}\left(\omega ,V_{\mathrm{ph}}\left(\omega \right)\right)-S_{\pi }\left(\omega \right)}{1-S_{\pi }\left(\omega \right)},$ wherein ω is frequency, V.sub.ph(ω) is phase-velocity, S.sub.NLSC.sup.ij is a normalized dispersion map using ith and jth traces, σ is a nonnegative parameter to control resolution, S.sub.NL.sup.ij is an unnormalized dispersion map represented by: $S_{\mathrm{NL}}^{\mathrm{ij}}\left(\omega ,V_{\mathrm{ph}}\right)=\frac{1}{T}\underset{0}{\stackrel{T}{\int }}\exp \left(-\frac{{\left[{\stackrel{\_}{d}}_i\left(t;\omega \right)-{\stackrel{\_}{d}}_j\left(t+\frac{x_{\mathrm{ij}}}{V_{\mathrm{ph}}};\omega \right)\right]}^2}{4{\omega }^2{}^{-2}{\sigma }^2}\right)\mathrm{dt},$ wherein T is the length of the measured time window, d.sub.i and d.sub.j are recorded seismic traces normalized by variance and represented as: ${\stackrel{\_}{d}}_i\left(t;\omega \right)={\sigma }_i^{-1}{d}_i\left(t;\omega \right),{\stackrel{\_}{d}}_j\left(t+\frac{x_{\mathrm{ij}}}{V_{\mathrm{ph}}};\omega \right)={\sigma }_j^{-1}{d}_j\left(t+\frac{x_{\mathrm{ij}}}{V_{\mathrm{ph}}};\omega \right),$ wherein σ.sub.i and σ.sub.j are variances defined as: ${\sigma }_i^2={\int }_0^T{d}_i^2\left(t;\omega \right)\mathrm{dt},{\sigma }_j^2{\int }_0^T{d}_j^2\left(t+\frac{x_{\mathrm{ij}}}{V_{\mathrm{ph}}};\omega \right)\mathrm{dt},$ and S.sub.π is a reference value for normalization represented by:
S.sub.π(ω)=I.sub.0(b)e.sup.−b, wherein $b=\frac{{}^2}{{\sigma }^2{\omega }^{2}T}$ and I.sub.0 is a modified Bessel function of zero-th order.

3. The method of claim 1, wherein the step of generating predicted surface waves for each receiver comprises using the following equation: $u_{\mathrm{surf}}^{\mathrm{pred}}\left(x;\omega \right)=\frac{1}{A}\underset{{\mathrm{dx}}_i=-L}{\overset{L}{.Math.}}{a}_{i}u\left(x+{\mathrm{dx}}_i;\omega \right)\exp \left[\mathrm{i\omega }\frac{{\mathrm{dx}}_i}{v_{\mathrm{ph}}\left(\omega \right)}\right],$ wherein u.sub.surf.sup.pred(x;ω) is the predicted surface wave at a receiver located at x in frequency ω domain, $A=\underset{{\mathrm{dx}}_i=-L}{\overset{L}{.Math.}}{a}_i,$ wherein a.sub.i is a weighting factor and L is a local spatial window size around x, u(x+dx.sub.i;ω) is a recorded seismic trace at location x+dx.sub.i including surface waves and body waves, $\exp \left[\mathrm{i\omega }\frac{{\mathrm{dx}}_i}{v_{\mathrm{ph}}\left(\omega \right)}\right]$ is a phase shift operator to correct for surface wave propagation effect, and v.sub.ph(ω) is estimated surface wave phase velocity generated using nonlinear signal comparison (NLSC).

4. A method for predicting surface waves in seismic data to facilitate their removal from the seismic data, comprising: obtaining and recording input multi-channel seismic data from a plurality of receivers, wherein the input multi-channel seismic data comprises a plurality of recorded seismic traces at a plurality of locations near the receivers; generating estimated surface wave phase velocities from the input multi-channel seismic data using nonlinear signal comparison (NLSC); and generating predicted surface waves for each receiver by using the estimated surface wave phase velocities and performing a phase shift and local stacking analysis for seismic traces at locations near each receiver.

5. The method of claim 4, wherein the step of generating estimated surface wave phase velocities comprises using the following equation: $S_{\mathrm{NLSC}}^{\mathrm{ij}}\left(\omega ,V_{\mathrm{ph}};\sigma \right)=\frac{S_{\mathrm{NL}}^{\mathrm{ij}}\left(\omega ,V_{\mathrm{ph}}\left(\omega \right)\right)-S_{}\left(\omega \right)}{1-S_{}\left(\omega \right)},$ wherein ω is frequency, V.sub.ph(ω) is phase-velocity, S.sub.NLSC.sup.ij is a normalized dispersion map using ith and jth traces, σ is a nonnegative parameter to control resolution, S.sub.NL.sup.ij is an unnormalized dispersion map represented by: $S_{\mathrm{NL}}^{\mathrm{ij}}\left(\omega ,V_{\mathrm{ph}}\right)=\frac{1}{T}\underset{0}{\stackrel{T}{\int }}\exp \left(-\frac{{\left[{\stackrel{\_}{d}}_i\left(t;\omega \right)-{\stackrel{\_}{d}}_j\left(t+\frac{x_{\mathrm{ij}}}{V_{\mathrm{ph}}};\omega \right)\right]}^2}{4{\omega }^2{}^{-2}{\sigma }^2}\right)\mathrm{dt},$ wherein T is the length of the measured time window, d.sub.i and d.sub.j are recorded seismic traces normalized by variance and represented as: ${\stackrel{\_}{d}}_i\left(t;\omega \right)={\sigma }_i^{-1}{d}_i\left(t;\omega \right),{\stackrel{\_}{d}}_j\left(t+\frac{x_{\mathrm{ij}}}{V_{\mathrm{ph}}};\omega \right)={\sigma }_j^{-1}{d}_j\left(t+\frac{x_{\mathrm{ij}}}{V_{\mathrm{ph}}};\omega \right),$ wherein σ.sub.i and σ.sub.j are variances defined as: ${\sigma }_i^2={\int }_0^T{d}_i^2\left(t;\omega \right)\mathrm{dt},{\sigma }_j^2{\int }_0^T{d}_j^2\left(t+\frac{x_{\mathrm{ij}}}{V_{\mathrm{ph}}};\omega \right)\mathrm{dt},$ and S.sub.π is a reference value for normalization represented by:
S.sub.π(ω)=I.sub.0(b)e.sup.−b, wherein $b=\frac{{}^2}{{\sigma }^2{\omega }^{2}T}$ and I.sub.0 is a modified Bessel function of zero-th order.

6. The method of claim 4, wherein the step of generating predicted surface waves for each receiver comprises using the following equation: $u_{\mathrm{surf}}^{\mathrm{pred}}\left(x;\omega \right)=\frac{1}{A}\underset{{\mathrm{dx}}_i=-L}{\overset{L}{.Math.}}{a}_{i}u\left(x+{\mathrm{dx}}_i;\omega \right)\exp \left[\mathrm{i\omega }\frac{{\mathrm{dx}}_i}{v_{\mathrm{ph}}\left(\omega \right)}\right],$ wherein u.sub.surf.sup.pred(x;ω) is the predicted surface wave at a receiver located at x in frequency ω domain, $A=\underset{{\mathrm{dx}}_i=-L}{\overset{L}{.Math.}}{a}_i,$ wherein a.sub.i is a weighting factor and L is a local spatial window size around x, u(x+dx.sub.i;ω) is a recorded seismic trace at location x+dx.sub.i including surface waves and body waves, $\exp \left[\mathrm{i\omega }\frac{{\mathrm{dx}}_i}{v_{\mathrm{ph}}\left(\omega \right)}\right]$ is a phase shift operator to correct for surface wave propagation effect, and v.sub.ph(ω) is estimated surface wave phase velocity generated using nonlinear signal comparison (NLSC).

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0013] FIG. 1 shows (a) a Multi-layered model with Vp (P-wave velocity), Vs (shear wave velocity) and density, (b) recorded vertical-component displacements of the multimode Rayleigh waves, (c) a dispersion map created using Nonlinear signal comparison (NLSC), and (d) a comparison of the original input surface waves and predicted surface waves.

[0014] FIG. 2 shows (a) a multi-layered model with Vp, Vs and density, (b) a full-wave elastic synthetic shot gather containing surface waves and other types of waves, (c) predicted surface waves including the fundamental mode and first overtone Rayleigh waves, and (d) the shot gather after removing the surface waves.

[0015] FIG. 3 shows (a) a comparison of predicted surface waves and modeled surface waves for fundamental mode Rayleigh wave, (b) a comparison of predicted surface waves and theoretical surface waves for first overtone Rayleigh wave, and (c) a dispersion map created using NLSC.

[0016] FIG. 4 shows (a) a recorded raw seismic shot gather, (b) the shot gather after removing the surface waves, (c) the location of source and receivers, and (d) the offset for each trace, indicating a gap in the distribution of receivers.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

[0017] The present disclosure relates to methods for processing seismic data to remove surface waves. Preferred embodiments include a method to predict and separate dispersive surface waves based on dispersion estimation that is completely data-driven. Nonlinear signal comparison (NLSC) is used to obtain a high resolution and accurate dispersion. Then based on the dispersion, surface waves are predicted from the input data using phase shift. The predicted surface waves are then subtracted from the original data.

[0018] Preferred embodiments described herein include the use of a dispersion measurement based on NLSC to estimate frequency-dependent phase velocities from seismic data. The dispersion measurement considers two time (t)-domain seismic traces, d.sub.i(t) and d.sub.j(t), recorded by two geophones, the i-th and the j-th geophones. The distance between the two geophones is X.sub.ij. The high-resolution dispersion map is obtained based on the nonlinear signal comparison (NLSC) described as:

[00001]$S_{\mathrm{NLSC}}^{\mathrm{ij}}\left(\omega ,V_{\mathrm{ph}};\sigma \right)=\frac{S_{\mathrm{NL}}^{\mathrm{ij}}\left(\omega ,V_{\mathrm{ph}}\left(\omega \right)\right)-S_{\pi }\left(\omega \right)}{1-S_{\pi }\left(\omega \right)}$

where ω and V.sub.ph(ω) are the frequency and phase-velocity, respectively; S.sub.NLSC.sup.ij is the normalized dispersion map using the ith and jth traces; σ is a nonnegative parameter to control the resolution. As σ.fwdarw.∞, the NLSC becomes the traditional crosscorrelation. In the above equation, S.sub.NL.sup.ij and S.sub.π are the unnormalized dispersion map and the reference value for normalization, respectively. They can be represented as:

[00002]$S_{\mathrm{NL}}^{\mathrm{ij}}\left(\omega ,V_{\mathrm{ph}}\right)=\frac{1}{T}\stackrel{T}{\underset{0}{\int }}\exp \left(-\frac{{\left[{\stackrel{\_}{d}}_i\left(t;\omega \right)-{\stackrel{\_}{d}}_j\left(t+\frac{x_{\mathrm{ij}}}{V_{\mathrm{ph}}};\omega \right)\right]}^2}{4{\omega }^2{\pi }^{-2}{\sigma }^2}\right)\mathrm{dt}$$\mathrm{and}$$S_{\pi }\left(\omega \right)=I_0\left(b\right)e^{-b},b=\frac{{\pi }^2}{{\sigma }^2{\omega }^{2}T},$

where I.sub.0 is the modified Bessel function of the zero-th order. d.sub.i and d.sub.j are two traces normalized by its variance, which can be represented as:

[00003]${\stackrel{\_}{d}}_i\left(t;\omega \right)={\sigma }_i^{-1}{d}_i\left(t;\omega \right),{\stackrel{\_}{d}}_j\left(t+\frac{x_{\mathrm{ij}}}{V_{\mathrm{ph}}};\omega \right)={\sigma }_j^{-1}{d}_j\left(t+\frac{x_{\mathrm{ij}}}{V_{\mathrm{ph}}};\omega \right)$

where σ.sub.i and σ.sub.j are the variances of the data defined as:

[00004]${\sigma }_i^2={\int }_0^T{d}_i^2\left(t;\omega \right)\mathrm{dt},{\sigma }_j^2={\int }_0^T{d}_j^2\left(t+\frac{x_{\mathrm{ij}}}{V_{\mathrm{ph}}};\omega \right)\mathrm{dt}$

where T is the length of the measured time window.

[0019] From the first equation above, the S.sub.NLSC.sup.ij range is from 0 to 1. Under the special case σ.fwdarw.∞. S.sub.NLSC.sup.ij will reduce to the traditional S.sub.LSC.sup.ij. The present S.sub.NLSC.sup.ij has a uniform resolution over a wide band of frequencies and the resolution can be controlled by a single parameter σ. To apply the above dispersion analysis on the active surface seismic data with multiple channels, S.sub.NLSC.sup.ij is averaged from all possible pairs of receivers to obtain the final dispersion map.

[0020] Importantly, the present methods produce uniform high resolution dispersion at both low and high frequencies. The traditional cross-correlation based dispersion measurement technique is a special use of the NLSC method. The NLSC method allows for the estimation of phase velocities using algorithms by first picking the local maximum at each frequency. Once the phase velocities have been picked, the surface waves can be estimated at each receiver location using phase shift and local stacking. For simplification, it is assumed that the receivers are distributed along a line in the x-direction. The surface wave is predicted using:

[00005]$u_{\mathrm{surf}}^{\mathrm{pred}}\left(x;\omega \right)=\frac{1}{A}\underset{{\mathrm{dx}}_i=-L}{\overset{L}{.Math.}}{a}_{i}u\left(x+{\mathrm{dx}}_i;\omega \right)\exp \left[i\omega \frac{{\mathrm{dx}}_i}{v_{\mathrm{ph}}\left(\omega \right)}\right],A=\underset{{\mathrm{dx}}_i=-L}{\overset{L}{.Math.}}{a}_i,$

where u.sub.surf.sup.pred(x;ω) is the predicted surface wave at the receiver located at x in the frequency ω domain; u(x+dx.sub.i;ω) is the recorded seismic trace at location x+dx.sub.i which includes both surface waves and body waves; L is a local spatial window size around x;

[00006]$\exp \left[i\omega \frac{{\mathrm{dx}}_i}{v_{\mathrm{ph}}\left(\omega \right)}\right]$

is the phase shift operator to correct for surface wave propagation effect; v.sub.ph(ω) is the estimated phase velocity from NLSC dispersion measurement; and a.sub.i is a weighting factor that can be referred as the local wave reconstruction operation. Using the equation above, the surface waves can be predicted at each receiver location using its neighboring traces. Finally, the predicted surface waves are subtracted from the original data.

[0021] Preferred embodiments of the data-driven surface-wave removal method include three steps. First, extract surface wave phase velocities using NLSC technique. Second, predict the surface waves at each receiver location using the estimated phase velocities and neighboring traces from the original seismic data. Third, subtract the predicted surface waves from the original data.

Example 1

[0022] This example utilized a synthetic seismic shot gather containing only surface waves. In this example, the near-surface velocity model (Xia et al. 1999) was used to show the performance of embodiments of the present method in the prediction of surface waves. FIG. 1(a) shows a Multi-layered model with Vp (P-wave velocity), Vs (shear wave velocity) and density. The synthetic seismic shot gather was generated using the method by Herrmann (2013), which allows for modeling of only surface waves. FIG. 1(b) shows recorded vertical-component displacements of the multimode Rayleigh waves. There were 46 traces recorded on the surface every 2 m between offsets from 10 m to 100 m. The source was an explosive source, located at a horizontal distance of 0 m and at a depth of 10 m. The recording time was 1.024 s, sampled at 1 ms. The frequency band was from 3 Hz to 70 Hz. The modeled seismic data contained multimode surface waves, including the fundamental mode and overtones. The dispersion analysis was performed using the NLSC. FIG. 1(c) shows the dispersion map using the traces in FIG. 1(b) via the NLSC (with a=0.01). The three dash lines are the theoretical dispersion curves for the fundamental mode (mode 0), the first overtone (mode 1), and the second overtone (mode 2), respectively. The black dot at each frequency shows the picked local maximum of the dispersion map, which indicates the phase velocity. The black bar shows the range of points with values above 95% of the picked local maximum. From the dispersion map (FIG. 1(c)), phase velocities can be picked at different frequencies. Based on the measured phase velocities from the NLSC dispersion map, the surface waves were predicted using the equation above. The predicted surface waves were in excellent agreement with the recorded surface waves in both phase and amplitude. FIG. 1(d) shows a comparison of the original input surface waves and predicted surface waves. From this example, it was observed that that the present methods can predict the dispersive surface waves, including multiple modes, by using only the recorded data.

Example 2

[0023] This example utilized a synthetic seismic shot gather using elastic full wavefield. The second synthetic data was modeled using the spectral element method (SEM) (e.g., Komatitsch and Vilotte 1998, Komatitsch and Tromp 2002) by solving the full elastic wave equation. The computational model and shot gathers are shown in FIGS. 2(a) and 2(b), respectively. FIG. 2(a) shows a multi-layered model with Vp, Vs and density. At shallow depths, there were several thin layers to approximate a velocity gradient. At depths of 500 m and 1000 m, two strong reflective interfaces exist. In FIG. 2(b), both surface waves as well as reflected/refracted/converted waves are observed. FIG. 2(b) shows a full-wave elastic synthetic shot gather containing surface waves and other types of waves. FIG. 3(c) shows a dispersion map of the traces in FIG. 2(b) via the NLSC (with σ=0.002). The two solid lines are theoretical dispersion curves for the fundamental mode (mode 0) and the first overtone (mode 1), respectively. FIG. 2(c) shows the predicted surface waves including the fundamental mode and first overtone Rayleigh waves. The surface waves in FIG. 2(c) were predicted by using dispersion measured for both the fundamental mode and the first overtone (Zheng and Hu, 2017) then following the procedure described above. FIG. 2(d) shows the shot gather after removing the surface waves. There were 101 surface receivers from offset 500 m to 1500 m at 10 m spacing and a depth of 10 m. The recording time length was 5 s. The time sampling interval was 1 ms. The source was an explosive source with a 15 Hz Ricker wavelet, located at a horizontal distance of 300 m and a depth of 30 m. Two circles in FIGS. 2(b) and 2(d) indicate where surface waves interfered with reflections for comparison. After removing the predicted surface waves (as in FIG. 2(d)), it is noted that reflection signals were not damaged by the present method, even at places where the reflections interfered with surface waves. Reflections become more visible after surface wave removal, which is noted by comparing the waveforms within the red circles in FIGS. 2(b) and 2(d).

[0024] To verify the fidelity of the predicted surface waves, surface waves were modeled for the fundamental mode and first overtone using the method by Herrmann (2013) and the results were compared to the surface waves predicted using the current methods. FIG. 3(a) shows a comparison of predicted surface waves and modeled surface waves for fundamental mode Rayleigh wave. FIG. 3(b) shows a comparison of predicted surface waves and theoretical surface waves for first overtone Rayleigh wave. From FIGS. 3(a) and 3(b), it is noted that predicted surface waves well match the modeled surface waves in both the fundamental mode and the first overtone. From this full elastic synthetic data test, it is noted that the present method can correctly predict the surface waves from data containing other types of waves. Furthermore, the present method can remove surface waves but without damaging useful signals.

Example 3

[0025] This example utilized field data, namely a field shot gather from a land acquisition survey. FIG. 4(a) shows the recorded raw seismic shot gather. FIG. 4(b) shows the shot gather after removing the surface waves. There were 35 traces with offsets from 356 m to 1529 m on the surface. The recording time was 1.5 s and the time sampling interval was 2 ms. The red ellipse indicates regions where surface waves interfered with the reflections. The distribution of trace has a gap at 17th trace. There was a lateral discontinuity in the surface wave moveout resulting from the irregular distribution of geophones. FIG. 4(c) shows the location of source and receivers. FIG. 4(d) shows the offset for each trace, indicating a gap in the distribution of receivers. FIG. 4(b) shows that the surface waves have been effectively removed (e.g., data within the circle). The surface wave removal algorithm can handle the irregularity because the surface wave removal method is locally performed. Because this method can effectively remove surface waves from field data with irregular geometry, it provides many potential benefits for processing real data.

[0026] The examples above show the successful application of the data-driven surface wave removal approach on three datasets, including two synthetic shot gathers and one field shot gather. All of these examples show that the present method is capable of predicting and suppressing surface waves from the data without damaging the reflections.

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