Frequency-domain wave-equation traveltime inversion with a monofrequency component

Wave-equation traveltime inversion (WTI) is a useful tool for background velocity model building. It is generally formulated and implemented in the time domain, in which the gradient is calculated by temporally crosscorrelating the source- and receiver-side wavefields. The time-domain source-side snapshots are either stored in memory or are reconstructed through back propagation. The memory requirements and computational cost of WTI are thus prohibitively expensive, especially for 3D applications. To partially alleviate this problem, we provide an implementation of WTI in the frequency domain with a monofrequency component. Because only one frequency is used, it is affordable to directly store the source- and receiver-side wavefields in memory. There is no need for wavefield reconstruction during gradient calculation. In such a way, we have dramatically reduced the memory requirements and computational cost compared with the traditional time-domain WTI realization. For practical implementation, the frequency-domain wavefield is calculated by time-domain finite-difference forward modeling and is transformed to the frequency domain by an on-the-fly discrete Fourier transform. Numerical examples on a simple lateral periodic velocity model and the Marmousi model demonstrate that our method can obtain accurate background velocity models comparable with those from time-domain WTI and frequency-domain WTI with multiple frequencies. A field data set test indicates that our method obtains a background velocity model that well predicts the seismic wave traveltime.

Towards autonomous event identifications in wave-equation traveltime inversion

We present a method to automatically relate events between observed and synthetic data for wave-equation traveltime (WT) inversion. The scheme starts with local similarity measurements by applying cross-correlation to localized traces. We then develop a differentiable alternative of the argmax function. By applying the differentiable argmax to each time slice of the local similarity map, we build a traveltime difference function but keep this process differentiable. WT inversion requires only the traveltime difference of related events through a phase shift. Thus, we must reject events that are not apparently related between observed and synthetic data. The local similarity map implies the possibility of doing so but shows abrupt changes with time and offset. To mitigate this problem, we introduce a dynamic programming algorithm to define a warping function. Wave packets between observed and synthetic data are assumed to be related if they are connected by this warping function, and they also exhibit high local similarity. Only such related events are considered in the subsequent calculation of misfit and adjoint source. Numerical examples demonstrate that the proposed method successfully retrieves an informative model from roughly selected data. In contrast, WT inversion based on cross-correlation or deconvolution fails to do so.

Joint wave-equation traveltime inversion of diving/direct and reflected waves for the P- and S-wave velocity macromodel building

Full waveform inversion (FWI) suffers from the local minima problem and requires a sufficiently accurate starting model to converge to the correct solution. Wave-equation traveltime inversion (WETI) is an effective tool to retrieve the long-wavelength components of the velocity model. We develop a joint diving/direct and reflected wave WETI (JDRWETI) method to build the P- and S-wave velocity macromodels. We estimate the traveltime shifts of seismic events (diving/direct waves, PP and PS reflections) through the dynamic warping scheme and construct a misfit function using both the time shifts of diving/direct and reflected waves. We derive the adjoint wave equations and the gradients with respect to the background models based on the joint misfit function. We apply the kernel decomposition scheme to extract the kernel of the diving/direct wave and the tomography kernels of PP and PS reflections. For an explosive source, the kernels of diving/direct wave and PP reflections and the kernel of PS reflections are used to compute the P- and S-wave gradients of the background models, respectively. We implement JDRWETI by a two-stage inversion workflow: first invert the P- and S-wave velocity models using the P-wave gradients and then improve the S-wave velocity model using the S-wave gradients. Numerical tests on synthetic and field datasets reveal that the JDRWETI method successfully recovers the long-wavelength components of P- and S-wave velocity models, which can be used for an initial model for the subsequent elastic FWI. Moreover, the proposed JDRWETI method prevails over the existing reflection WETI method and the cascaded diving/direct and reflected wave WETI method, especially when large velocity errors are present in the shallow part of the starting models. The JDRWETI method with the two-stage inversion workflow can give rise to reasonable inversion results even for the model with different P- and S-wave velocity structures.

Elastic reflection traveltime inversion with pure-wave propagators

Elastic full-waveform inversion (EFWI) updates high-resolution model parameters by minimizing the misfit function between the observed and modeled data. EFWI possesses strong nonlinearity and is likely to converge to a local minimum when the inversion begins with inaccurate initial models. Elastic reflection waveform inversion (ERWI) recovers the low-wavenumber components of P- and S-wave velocities along the "rabbit ear" wave paths to provide initial velocity models for EFWI. However, every iteration of ERWI requires six times as many forward calculations with elastic-wave equations which can be computationally expensive. Hence, we have developed a pure-wave reflection waveform inversion (PRWI) approach, which sequentially inverts low-wavenumber components of P- and S-wave velocity models. In our PRWI, we decompose elastic-wave operators into background and perturbed pure-wave parts and derive PRWI gradients using pure-wave operators. Both the background and perturbed wavefields in PRWI gradients are vector wavefields with single wave mode. PRWI can remove the high-wavenumber noise caused by S-wave stress decomposition, and reduce the computational cost of ERWI by almost 70%. Under the framework of PRWI, we have further developed the pure-wave reflection traveltime inversion (PRTI) approach to alleviate the issue of cycle skipping caused by waveform mismatch. In order to ensure the recovery of low-wavenumber components, we mute out the contribution of wavefields with small opening angles to PRTI gradients. Numerical examples have demonstrated that our PRTI method can provide good initial velocity models for EFWI efficiently.

Improved interpretation of SAGEEP 2011 blind refraction data using Frequency-Dependent Traveltime Tomography

<p>We demonstrate improved resolution in P-wave velocity tomograms obtained by inversion of the synthetic SAGEEP 2011 refraction traveltime data (Zelt 2010) using Wavepath-Eikonal Traveltime Inversion (WET; Schuster 1993) and Wavelength-Dependent Velocity Smoothing (WDVS; Zelt and Chen 2016). We use a multiscale inversion approach and a Conjugate-Gradient based search method. Our default starting model is a 1D-gradient model obtained directly from the traveltime first arrivals assuming diving waves (Sheehan, 2005). As a second approach, we map the first breaks to assumed refractors and obtain a layered starting model using the Plus-Minus refraction method (Hagedoorn, 1959). We compare tomograms obtained using WDVS to smooth the current velocity model grid before forward modeling traveltimes vs. tomograms obtained without WDVS. Results show that WET images velocity layer boundaries more sharply when engaging WDVS. We determine the optimum WDVS frequency iteratively by trial-and-error. We observe that the lower the used WDVS frequency, the stronger the imaged velocity contrast at the top-of-basement. Using a WDVS frequency that is too low makes WDVS based WET inversion unstable exhibiting increasing RMS error, too high modeled velocity contrast and too shallow imaged top-of-basement. To speed up WDVS, we regard each nth node only when scanning the velocity along straight scan lines radiating from the current velocity grid node. Scanned velocities are weighted with a Cosine-Squared function as described by (Zelt and Chen, 2016). We observe that activating WDVS allows decreasing WET regularization (smoothing and damping) to a higher degree than without WDVS.</p><p>References:</p><p><span>Hagedoorn, J.G., 1959, </span><span>The Plus-Minus method of interpreting seismic refraction sections, Geophysical Prospecting</span><span>, Volume 7, 158-182.</span></p><p><span>Rohdewald, S.R.C., 2021, SAGEEP11 data interpretation, https://rayfract.com/tutorials/sageep11_16.pdf.</span></p><p>Schuster, G.T., Quintus-Bosz, A., 1993, <span>Wavepath eikonal traveltime inversion: Theory</span>. Geophysics, Volume 58, 1314-1323.</p><p><span>Sheehan, J.R., Doll, W.E., Mandell, W., 2005, </span><span>An evaluation of methods and available software for seismic refraction tomography analysis</span><span>, JEEG, Volume 10(1), 21-34.</span></p><p>Shewchuk, J.R., 1994, An Introduction to the Conjugate Gradient Method Without the Agonizing Pain, <span>http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf</span><span>. </span></p><p>Zelt, C.A., 2010, Seismic refraction shootout: blind test of methods for obtaining velocity models from first-arrival travel times, <span>http://terra.rice.edu/department/faculty/zelt/sageep2011</span>.</p><p><span>Zelt, C.A., Haines, S., Powers, M.H. et al. 2013, </span><span>Blind Test of Methods for Obtaining 2-D Near-Surface Seismic Velocity Models from First-Arrival Traveltimes</span><span>, JEEG, Volume 18(3), 183-194. </span></p><p><span>Zelt, C.A., Chen, J., 2016, </span><span>Frequency-dependent traveltime tomography for near-surface seismic refraction data</span><span>, Geophys. J. Int., Volume 207, 72-88. </span></p>

First-arrival traveltime inversion of seismic diving waves observed on undulant surface

SUMMARY A non-iterative first-arrival traveltime inversion method (NFTI) is proposed for building smooth velocity models using seismic diving waves observed on irregular surface. The new ray and traveltime equations of diving waves propagating in smooth media with undulant observation surface are deduced. According to the proposed ray and traveltime equations, an analytical formula for determining the location of the diving-wave turning points is then derived. Taking the influence of rough topography on first-arrival traveltimes into account, the new equations for calculating the velocities at turning points are established. Based on these equations, a method is proposed to construct subsurface velocity models from the observation surface downward to the bottom using the first-arrival traveltimes in common offset gathers. Tests on smooth velocity models with rugged topography verify the validity of the established equations, and the superiority of the proposed NFTI. The limitation of the proposed method is shown by an abruptly-varying velocity model example. Finally, the NFTI is applied to solve the static correction problem of the field seismic data acquired in a mountain area in the western China. The results confirm the effectivity of the proposed NFTI.

A comparison of ray-based traveltime and waveform inversion results for depth velocity model building in complex fold and thrust belts

In onshore fold and thrust belt geology, the seismic wave propagation leads to complex waveforms due to large lateral variations in the earth parameters and multiple scattering effects. Moreover, due to the acquisition difficulty in mountainous terrain, the data coverage can be limited. Though waveform inversion should help in imaging in this context, finding a good-enough initial model parameter is challenging. Efficient ray-based traveltime inversions could be used when we focus on the early arrivals. However, they rely on a high-frequency assumption that may not be justified in this context. Indeed, significant interferences may occur within the first Fresnel zone. To evaluate the potential challenges, we invert both 3D passive and 2D active data sets recorded over a fold and thrust belt region in Albania. We compare ray-based high-frequency traveltime and finite-frequency waveform inversion results obtained by inverting local earthquake events and first arrivals from 2D active seismic data. We consider only acoustic propagation. The two results differ significantly. The ray-based traveltime inversion we used does not always give a velocity model that can be used as an initial model for waveform inversion. This could be due to a bias toward the initial model when the data coverage is limited, since the inversions have non-zero null-space. It could also be a limitation of the ray-based traveltime inversion that assumes the phases of the picked events are linear in frequency over the considered frequency band.