Application of Weighted Early-Arrival Waveform Inversion to Shallow Land Data

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1 Application of Weighted Early-Arrival Waveform Inversion to Shallow Land Data Han Yu, Dongliang Zhang and Xin Wang King Abdullah University of Science and Technology, Division of Physical Science and Engineering, Thuwal Saudi Arabia ABSTRACT Recent studies have shown that inverting traces weighted by the energy of the early-arrivals can improve the accuracy of estimating shallow velocities. This is explained by showing that the associated misfit gradient function tends to be sensitive to the kinematics of wave propagation and insensitive to the dynamics. A synthetic example verifies the theoretical predictions and show that the effects of noise and unpredicted amplitude variations in the inversion are reduced using this weighted early arrival waveform inversion (WEWI). We also apply this method to a D land data set for estimating the near-surface velocity distribution. The reverse time migration images suggest that, compared to the tomogram inverted directly from the early arrival waveforms, the WEWI tomogram provides a more convincing velocity model and more focused reflections in the deeper part of the image.. INTRODUCTION The near-surface velocity distribution is crucial for imaging the deeper parts of the Earth. Complex velocity variations at the near surface are often associated with undulating topography or irregular geology in the near-surface weathered layers (Amorium et al., 987; Taner et al., 998). If the near-surface velocity distribution is not accurately estimated, the coherency of the deeper migrated reflections can be strongly degraded (White, 989; Marsden, 993). To partly remedy this problem, the near-surface velocity model with smooth variations can be estimated by traveltime tomography (Zhu and McMechan, 989; Pratt and Goulty, 99; Aki and Richards, 00) that inverts the first-arrival traveltimes. However, in geologically complex areas, a more highly resolved velocity model is needed for imaging deeper reflectors. In this regard, waveform inversion (Tarantola, 984; Mora, 987; Zhou et al., 995) was developed to invert for more accurate tomograms by finite-frequency 6 seismic wave propagation. To reduce the computational time and local minima problems (Sirgue and Pratt, 004), early-arrival waveform inversion (EWI) was proposed by Sheng et al. (006) in the space-time domain and later applied to marine data (Boonyasiriwat et al., 00). In this work, we carry out the inversion on land data by following the conventional EWI method but using a recently developed objective misfit function (Shen, 00), whose gradient is more robust and focuses more on matching the phase rather than the amplitude in the data. However, the associated gradient does not have an important energy normalization term which is important for optimal imaging. In this work, the gradient associated with this weighted early arrival waveform inversion (WEWI) is properly normalized and shown to significantly improve the accuracy of the final tomogram. Instead of replacing the amplitude spectrum of a calculated trace with that of the corresponding observed trace (Sun and Schuster, 993), we implement WEWI in the time domain by normalizing both the observed and calculated early arrivals using the L norm of the trace, where this approach avoids the phase wrapping problem in the frequency domain (Shin and Min, 006). Our synthetic results demonstrate that compared to EWI, WEWI can mitigate the effects of noise and unpredicted amplitude variations in the data and robustly invert for highly resolved near-surface tomogram. Moreover, a land data test illustrates that WEWI produces a more accurate shallow subsurface tomogram where the energy is focused in the deeper part. This paper is organized into four sections. The first part is the introduction, and the second part analyzes the misfit function associated with its gradient in our approach. In section 3, numerical results are shown for inverting data associated with the Marmousi model and a field experiment in Saudi Arabia. The last section presents the conclusions.. THEORY In many field, particularly land data sets, there are strong elastic arrivals such as surface waves that cannot be modeled by the

2 6 Yu et al. acoustic wave equation. In addition, the amplitudes of some traces are distorted due to unexplained environmental sources and not explained by geometric spreading. Some elastic effects in the data can be reduced by applying an early arrival window to mute the later arrivals. Therefore, the conventional waveform inversion misfit function is modified by Shen (00) and expressed in the time domain as E= = p(x r, t x s ), s, r s, r p calc (x r, t x s ) p obs(x r, t x s ) p calc (x r, t x s ) p obs (x r, t x s ), () which indicates that p pt p calc p p calc calc eliminates the phase information of p calc in p obs. Therefore the common phases in p calc and p obs will not be selected to match again in the next iteration. Figure shows fact the fact this new virtual source is perpendicular to the recorded data. It thus weakens the effects of p calc and strengthens the phase difference between p obs and p calc in the back propagated wave fields, therefore making WEWI robust and significant. where p(x r, t x s ) denotes the pressure field trace recorded at the receiver position x r, with listening time t and a source at x s ; p denotes the L norm of the N vector p, namely p T p, where N is the number of time samples in the trace. Here, p obs represents the recorded trace with windowed early arrivals and p calc represents the synthetic early arrivals. The synthetic data are calculated by solving the constant-density acoustic wave equation, p(x, t x s ) c (x) t p(x, t x s )= s(x, t x s ), () where c(x) represents the velocity model at position x. The solution to equation is calculated by a second order in time and eighth order in space staggered-grid method (Levander, 988). Equation normalizes the observed and the synthetic early arrivals so that their energy can be compared at the same scale, and the waveform inversion in this case is more sensitive to phase differences in the misfit function. To unveil this fact, the Fréchet derivative [grad(x)= E c(x) ] of the functional E with respect to c(x) is calculated by grad(x)= p T (p calc/ p calc ) c = ( p pt p calc p calc p calc p calc ) T p calc c. (3) Here, if the scalar / p calc is ignored, the first term in equation 3 exactly corresponds to the gradient of conventional waveform misfit function. In this paper, we invert the field data using this gradient but ignore / p calc because it is included in calculating the step length when updating the velocity model. Previous synthetic tests (Shen, 00) missed the superscript term in the denominator p calc of equation 3, thus making the phase match less accurate. This misfit function becomes more significant if the term pt p calc is not small, which can be caused p calc by complicated geological conditions, because it attaches more importance to accurately predicting the phases rather than the amplitudes. Note, that the gradient term p pt p calc p p calc calc is orthogonal to p calc since ( p ( p)t p calc p calc p calc ) T p calc = p T p calc p T p calc = 0, (4) Figure : Construction of data residuals as back propagating sources for EWI and WEWI. The velocity model is estimated by an iterative conjugate gradient method where c k+ (x)=c k (x)+λ k d k (x), (5) and the conjugate directions are defined by d k = P k g k +β k d k, (6) for iterations k=,,..., k max, g=[grad(x)], and P is the conventional geometrical-spreading preconditioner (Causse et al., 999). The scalarλ k is the step length which can be determined by a quadratic line-search method (Nocedal and Wright, 999), and d k (x) is the component of the direction vector d k (x) indexed by x. For the first iteration, we set d 0 = g 0. The parameterβ k is calculated by the Polak-Ribiére formula (Nocedal and Wright, 999) β k = gt k (P kg k P k g k ) g T k P k g k. (7) To compute the gradient direction at each iteration reduces to computing the reverse time migration operation. Additional forward modelings are required for the line search. The initial velocity model c 0 (x) is the traveltime tomogram inverted by picked first arrivals (Nemeth et al., 997), and equation 5 is iteratively applied until the objective functional E satisfies a stopping criterion.

3 Weighted EWI to Shallow Land Data NUMERICAL TESTS OF WEWI 3. Synthetic Data Example The Marmousi model is used to test the robustness and quality of WEWI before it is applied to a land data set in the next section. First, a synthetic data set is generated based on the Marmousi model (Figure (a)) with a 576 (horizontal) by 84 (vertical) gridded mesh with a 6.0 m grid interval. There are 60 shots with a 54 m shot spacing, and for each shot, the number of receivers is 90 with a 6 m receiver spacing. The recording length is.5 s with a sampling rate 0.5 ms. White noise is added to each trace of every shot gather. The nonzero meanvalued noise consists of two parts: random receiver noise is added to every CSG and a random static amplitude shift is also applied to each trace. Figures 3(a) and 3(b) show one common shot gather before and after adding the noise, and they are displayed with the same amplitude scale. Figure 3(c) shows the noise mask added to Figure 3(a) that simulates dead traces. Finally, WEWI and conventional EWI are used to invert the noisy data up to 0.5 s after the first arrival using the same initial velocity model (see Figure (b)). The resulting tomograms after the 30 th iteration are presented in Figures 4(a) and 4(b), which proves that WEWI is less sensitive to the noise; the matched phase information in the predicted data is excluded in calculating the gradient using reverse time migration (RTM). Their misfit gradient Figures 5(a) and 5(b) for the first iteration further validate the advantage of the WEWI method, followed by two obviously different convergence rates of the data residual shown in Figure 5(c). The comparison of the two tomograms apparently indicates WEWI can invert for a more accurate velocity model with unpredicted noise in the data, and imply the robustness of the WEWI compared to EWI (a) Marmousi Velocity Model (b) Initial Velocity Model Figure : The Marmousi models with (a) the true velocity distribution and (b) the its smoothed version as the initial velocity model for waveform inversion. km/s (a) CSG #0 before Adding the Noise (b) CSG #0 after Adding the Noise (c) The Noise Mask for CSG # Figure 3: The CSG #0 generated by the Marmousi model (a) before, (b) after adding the noise, and (c) its noise mask. 3. Land Data Example 3.. Acquisition and Processing A D seismic survey is was carried out near KAUST with the acquisition geometry illustrated in Figure 6. The D acquisition line consists of 79 shots and 40 vertical component geophones per shot, with a uniform spacing of 30 m for both shots and receivers. For each channel, the record length is s with a 4 ms sampling rate. For a common shot gather (CSG), the shot position is in the middle of the 40 receivers, so the largest offset is 0 m. WEWI is applied to the first CSGs, and the horizontal distance for our inversion is restricted between shot # and # (red crosses in Figure 6). The topography of these shots is shown in Figure 7, which is almost flat if the horizontal distance X and the depth Z are of the same scale. In this case, we choose Z= 5 m as the surface for the inversion and ignore the elevation variations of the geophones and shotpoints. Prior to data processing, it is useful to estimate the nearsurface velocity distribution from the picked first arrivals. Figure 8 shows CSG #7 with picked first arrivals marked by red crosses, and the direct wave and the refractions are respectively marked by the white and green dashed lines. The slopes of the

4 64 Yu et al. (a) WEWI Tomogram (a) WEWI Gradient for the First Iteration km/s (b) Conventional EWI Tomogram (b) EWI Gradient for the First Iteration Figure 4: The inverted tomograms using (a) WEWI and (b) conventional EWI. two lines suggest that the velocity corresponding to the first layer is approximately v = 750 m/ s=3750 m/s, whereas the velocity for the second layer is about v = 800 m/0.5 s= 5600 m/s. We first apply the F-K filter to remove apparent surface waves in all the CSGs. As an example, the raw CSG # is shown in Figure 9(a), with most of the surface waves eliminated in Figure 9(b) by F-K filtering. The field data are then transformed from 3D to D format by applying the filter i/ω to all the traces in the frequency domain and scaled by t for 3D geometrical spreading in the time domain. In the meantime, the data are also filtered with a pass band from 0 to 0 Hz. The results after filtering of CSG # are shown in Figure 9(c). The spectra of a trace in CSG # before and after the processing steps are presented in Figure 0. Normalized L Data Residual (c) Data Residual Convergence Rate Comparison for Synthetic Data Iteration Number WEWI Convetional EWI Figure 5: (a) WEWI and (b) conventional EWI gradients for the first interation, and (c) the comparison of their convergence rate. Acquisition Geometry with Equal Scales in Distance 3.. Applying WEWI to The Land Data Set WEWI is implemented using a starting model calculated by first-arrival traveltime tomgoraphy with a 757 (horizontal) by 97 (vertical) gridded mesh and a 7.5 m grid interval. First, a ray-based traveltime tomography method is used to invert the first arrival traveltimes for a smooth velocity model. The traveltime tomogram is shown in Figure, where the high-velocity layer is about 80 m deep, as illustrated by the refraction data in Figure 8. Second, WEWI is used to invert the data restricted by dynamically increasing time windows. The time window stretches from the beginning of the record to 4 s after the first arrivals for the first 0 iterations and its size quadratically increases up to 0.5 s in the next 0 iterations. All the traces bounded by their time windows in the observed and calculated data are normalized according to equation during each iteration. The / p calc term in equation 3 is dropped and the rest of the terms are exactly followed when calculating the gradient. Y [km].6 0 Red Crosses: the first CSGs for inversion X [km] Figure 6: The acquisition geometry with equal scales in the horizontal and the vertical directions. The inverted WEWI tomogram at the 30 th iteration is shown in Figure (a). At the 0 th iteration, the observed data and cal-

5 Weighted EWI to Shallow Land Data 65 0 The Topography of Shot to Figure 7: Topography of the first shots.. (a) Raw CSG # CSG # 7 with Picked First Arrivals Figure 8: A common shot gather with its picked first arrivals. Red crosses: picked first arrivals; green dashed line: the estimated direct wave; white dash line: the estimated refraction. culated data recorded 4 s after the first arrivals mostly agree with one another in Figures 3(a) and 3(b). Note that nearoffset traces within 50 m from the source and far-offset traces more than 3300 m from the sources are not inverted due to the poor quality of the recorded data. After the dynamic windows are applied, many later arrival events also match according to Figures 4(a) and 4(b) at the 30 th iteration without harming the matched previous arrivals. Figure 5 depicts the residual vs. iteration number plot when applying WEWI for 30 iterations. Note that the residual gradually includes more data for comparison while the window size grows very slowly. To verify the inverted velocity model, we migrate the high frequency portion ( 45 Hz) of the observed data using the traveltime and the WEWI tomograms as migration velocity models. The reverse time migration (RTM) images above m in depth are exhibited in Figures 6(a) and 6(b) and their associated common image gathers are also shown in Figures 7(a) and 7(b). The RTM image based on the WEWI tomogram shows a more continuous structure in the shallow part and more focused energy in the deep part compared to the RTM image computed with the traveltime tomogram. The CIGs based on the WEWI tomogram also are flatter for both the near-surface and the deeper reflectors. The EWI is tomogram shown in Figure (b). Some shallow reflectors can still be detected according to this tomogram but they are much stronger compared to the tomogram in Figure (a). Figure (b) could be less realistic because the estimated near-surface velocity at Z = 80m is around 5600 m/s and not higher than 6000 m/s according to the picked first arrivals in Figure 8. The calculated CSG # for the degraded tomogram presented in Figure 3(c) does not match well with (b) F K Filtered CSG # (c) CSG # after Bandpass Filtering and 3D to D Transformation Figure 9: A raw (a) CSG #, (b) with its surface waves removed by the dip filter, and (c) the CSG # after bandpass filtering and 3D to D transformation. Figure 3(a) even for the early arrivals at the intermediate 0 th iteration. When the time window increases to 0.5 s in the 30 th iteration, the calculated CSG # with conventional EWI (4(c)) shows greater difference from the observed CSG # than its counterpart with WEWI (Figure 4(b)). The RTM image (Figure 6(c)) using this tomogram is also inferior compared to Figure 6(b), and so are the CIGs (Figures 7(c) and 8(c)) associated with it. Moreover, the normalized EWI data residual only decreases to 5 after 30 iterations as shown in Figure CONCLUSIONS A modified misfit function is applied and its associated Fréchet derivative is exactly followed to calculate the gradient for updating the velocity model. This modified function using traceby-trace normalization attaches more importance to accurately

6 66 Yu et al. (a) Spectrum of a Trace of Raw CSG # (a) Observed CSG # Normalized Amplitude Frequency (Hz) (b) Spectrum of the Trace in (a) after Filtering Normalized Amplitude Frequency (Hz) (b) Calculated CSG # with WEWI Figure 0: The spectrum of a trace in CSG # (a) before and (b) after processing. Traveltime Tomogram km/s (c) Calculated CSG # with Conventional EWI Figure : The traveltime tomogram for the picked first arrivals from CSGs # #. (a) WEWI Tomogram (b) Conventional EWI Tomogram Figure : (a) The WEWI, and (b) the conventional EWI tomograms inverted from the early arrivals of CSGs # #. predicting the phase than the amplitude of the recorded data. The gradient associated with this objective function mitigates km/s Figure 3: (a) The processed CSG #, (b) the calculated CSG # with the WEWI gradient, and (c) the calculated CSG # computed with conventional EWI. the effects of noise and unpredicted amplitude variations, and it strengthens the frequency difference between the observed and calculated data in the back propagated wave fields. The robustness and the phase matching property of WEWI are then validated by inverting for the Marmousi model using a polluted data set. This method is also tested on a real case by inverting CSGs of a land data set after regular processing. A dynamic time window is also used to invert the early arrivals. So that both refractions and some early arriving reflections are included in the WEWI approach. Our field data results suggest that WEWI can generate a more accurate and highly resolved velocity model compared to the conventional EWI tomogram. Although the data with peak frequency around 5 Hz are first used in the inversion, the drawbacks of WEWI for land data largely come from the lack of lower frequency data from -5 Hz. Part of this problem can come from the filters that remove low frequency information polluted by surface waves or other noise, and this might be remedied by better processing

7 Weighted EWI to Shallow Land Data 67 (a) Observed CSG # Muted by A Dynamic Time Window (a) RTM Image Using Traveltime Tomogram (b) RTM Image using WEWI Tomogram (b) Calculated CSG # Muted by A Dynamic Time Window (c) RTM Image Using Conventional EWI Tomogram (c) Calculated CSG # with Conventional EWI Figure 6: RTM images based on (a) the traveltime tomogram, (b) the WEWI tomogram, and (c) the conventional EWI tomogram. to continue widening the time window or inverting the events from deeper reflectors Figure 4: (a) The processed CSG #, (b) the calculated CSG # with WEWI, and (c) the calculated CSG # with EWI muted by a dynamic time window. Normalized L Data Residual 0.9 Data Residual Convergence Rate Comparison for Land Data WEWI Conventional EWI Iteration Nunber Figure 5: The convergence rate of WEWI for 30 iterations. techniques. Also, WEWI may still encounter the cycle skipping problem although it better utilizes the phase information in the data. Cycle skipping problems can be partly overcome by only inverting the near offset traces, and then with later iterations invert the larger offset traces. For reflections events from deep reservoir geology in oil industry, it is also necessary 5. ACKNOWLEDGMENTS We would like to thank the 03 sponsors of the CSIM Consortium ( for their financial support. The computation resource Shaheen ( for inversion provided by the high performance computing (HPC) center of King Abdullah University of Science and Technology (KAUST) is greatly appreciated. We also thank Prof. Schuster and anonymous CSIM members for their professional comments in the development of this work. REFERENCES Aki, K. and P. G. Richards, 00, Quantitative seismology: nd edition. University Science Books. Amorium, W. N. D., P. Hubral, and M. Tygel, 987, Computing field statics with the help of seismic tomography: Geophysical Prospecting, 35, Boonyasiriwat, C., G. T. Schuster, P. Valasek, and W. Cao, 00, Applications of multiscale waveform inversion to marine data using a flooding technique and dynamic earlyarrival windows: Geophysics, 75, R9 R36. Causse, E., R. Mittet, and B. Ursin, 999, Preconditioning for full-waveform inversion in viscoacoustic media: Geophysics, 64, Levander, A. R., 988, Fourth-order finite-difference p-sv seismograms: Geophysics, 53,

8 68 Yu et al. (a) CIGs based on Traveltime Tomogram (a) CIGs based on Traveltime Tomogram (b) CIGs based on WEWI Tomogram (b) CIGs based on WEWI Tomogram (c) CIGs based on Conventional EWI Tomogram (c) CIGs based on Conventional EWI Tomogram Figure 7: One zoomed view of common image gathers based on (a) the traveltime tomogram, (b) the WEWI tomogram, and (c) the conventional EWI tomogram. Marsden, D., 993, Statics corrections-a review: The Leading Edge,, Mora, P. R., 987, Nonlinear two-dimensional elastic inversion of multioff-set seismic data: Geophysics, 5, 8. Nemeth, T., E. Normark, and F. Qin, 997, Dynamic smoothing in crosswell traveltime tomography: Geophysics, 6, Nocedal, J. and S. J. Wright, 999, Numerical optimization: Springer series in operations research and financial engineering. Pratt, R. G. and N. R. Goulty, 99, Combining wave-equation imaging with traveltime tomography to form high-resolution images from crosshole data: Geophysics, 56, Shen, X., 00, Near-surface velocity estimation by weighted early-arrival waveform inversion: 80th SEG Technical Program Expanded Abstracts, Sheng, J., A. Leeds, M. Buddensiek, and G. T. Schuster, 006, Early arrival waveform tomography on near-surface refraction data: Geophysics, 7, U47 U57. Shin, C. and D. J. Min, 006, Waveform inversion using a logrithmic wavefield: Geophysics: Geophysics, 7, R3 R4. Figure 8: Another zoomed view of common image gathers based on (a) the traveltime tomogram, (b) the WEWI tomogram, and (c) the conventional EWI tomogram. Sirgue, L. and R. G. Pratt, 004, Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies: Geophysics, 69, Sun, Y. and G. T. Schuster, 993, Time-domain phase inversion: SEG Technical Program Expanded Abstracts 993, Taner, M. T., D. E. Wagner, E. Baysal, and L. Lu, 998, A unified method for -d and 3-d refraction statics: Geophysics, 63, Tarantola, A., 984, Inversion of seismic reflection data in the acoustic approximation: Geophysics, 49, White, D. J., 989, Two-dimensional seismic refraction tomography: Geophysics,, Zhou, C., W. Cai, Y. Luo, G. T. Schuster, and S. Hassanzadeh, 995, Acoustic wave-equation traveltime and waveform inversion of crosshole seismic data: Geophysics, 60, Zhu, X. and G. A. McMechan, 989, Estimation of a twodimensional seismic compressional-wave velocity distribution by iterative tomographic imaging: International Journal of Imaging System and Technology,, 3 7.