Conventional Waveform Inversion Research Articles

Multilayer Perceptron and Bayesian Neural Network-Based Elastic Implicit Full Waveform Inversion

We introduce and analyze the elastic implicit full waveform inversion (EIFWI) of seismic data, which uses neural networks to generate elastic models and perform full waveform inversion. EIFWI carries out inversion by linking two main networks: a neural network that generates elastic models and a recurrent neural network to perform the modeling. The approach is distinct from conventional waveform inversion in two key ways. First, it reduces reliance on accurate initial models relative to conventional FWI. Instead, it invokes general information about the target area, for instance, estimates of means and standard deviations of medium properties in the target area or, alternatively, well-log information in the target area. Second, iterative updating directly affects the weights in the neural network rather than the elastic model. Elastic models can be generated in the first part of the EIFWI process in either of two ways: through the use of a multilayer perceptron (MLP) network or a Bayesian neural network (BNN). Numerical testing is suggestive that the MLP-based EIFWI approach in principle builds accurate models in the absence of an explicit initial model, and the BNN-based EIFWI can give the uncertainty analysis for the prediction results.

Source‐independent waveform inversion for attenuation estimation in anisotropic media

ABSTRACTIn previous publications, we presented a waveform‐inversion algorithm for attenuation analysis in heterogeneous anisotropic media. However, waveform inversion requires an accurate estimate of the source wavelet, which is often difficult to obtain from field data. To address this problem, here we adopt a source‐independent waveform‐inversion algorithm that obviates the need for joint estimation of the source signal and attenuation coefficients. The key operations in that algorithm are the convolutions (1) of the observed wavefield with a reference trace from the modelled data and (2) of the modelled wavefield with a reference trace from the observed data. The influence of the source signature on attenuation estimation is mitigated by defining the objective function as the ℓ2‐norm of the difference between the two convolved data sets. The inversion gradients for the medium parameters are similar to those for conventional waveform‐inversion techniques, with the exception of the adjoint sources computed by convolution and cross‐correlation operations. To make the source‐independent inversion methodology more stable in the presence of velocity errors, we combine it with the local‐similarity technique. The proposed algorithm is validated using transmission tests for a homogeneous transversely isotropic model with a vertical symmetry axis that contains a Gaussian anomaly in the shear‐wave vertical attenuation coefficient. Then the method is applied to the inversion of reflection data for a modified transversely isotropic model from Hess. It should be noted that due to the increased nonlinearity of the inverse problem, the source‐independent algorithm requires a more accurate initial model to obtain inversion results comparable to those produced by conventional waveform inversion with the actual wavelet.

Unwrapped phase inversion with an exponential damping

Full-waveform inversion (FWI) suffers from the phase wrapping (cycle skipping) problem when the frequency of data is not low enough. Unless we obtain a good initial velocity model, the phase wrapping problem in FWI causes a result corresponding to a local minimum, usually far away from the true solution, especially at depth. Thus, we have developed an inversion algorithm based on a space-domain unwrapped phase, and we also used exponential damping to mitigate the nonlinearity associated with the reflections. We construct the 2D phase residual map, which usually contains the wrapping discontinuities, especially if the model is complex and the frequency is high. We then unwrap the phase map and remove these cycle-based jumps. However, if the phase map has several residues, the unwrapping process becomes very complicated. We apply a strong exponential damping to the wavefield to eliminate much of the residues in the phase map, thus making the unwrapping process simple. We finally invert the unwrapped phases using the back-propagation algorithm to calculate the gradient. We progressively reduce the damping factor to obtain a high-resolution image. Numerical examples determined that the unwrapped phase inversion with a strong exponential damping generated convergent long-wavelength updates without low-frequency information. This model can be used as a good starting model for a subsequent inversion with a reduced damping, eventually leading to conventional waveform inversion.

Waveform inversion with source encoding for breast sound speed reconstruction in ultrasound computed tomography.

Ultrasound computed tomography (USCT) holds great promise for improving the detection and management of breast cancer. Because they are based on the acoustic wave equation, waveform inversion-based reconstruction methods can produce images that possess improved spatial resolution properties over those produced by ray-based methods. However, waveform inversion methods are computationally demanding and have not been applied widely in USCT breast imaging. In this work, source encoding concepts are employed to develop an accelerated USCT reconstruction method that circumvents the large computational burden of conventional waveform inversion methods. This method, referred to as the waveform inversion with source encoding (WISE) method, encodes the measurement data using a random encoding vector and determines an estimate of the sound speed distribution by solving a stochastic optimization problem by use of a stochastic gradient descent algorithm. Both computer simulation and experimental phantom studies are conducted to demonstrate the use of the WISE method. The results suggest that the WISE method maintains the high spatial resolution of waveform inversion methods while significantly reducing the computational burden.

Image‐domain wavefield tomography with extended common‐image‐point gathers

ABSTRACTWaveform inversion is a velocity‐model‐building technique based on full waveforms as the input and seismic wavefields as the information carrier. Conventional waveform inversion is implemented in the data domain. However, similar techniques referred to as image‐domain wavefield tomography can be formulated in the image domain and use a seismic image as the input and seismic wavefields as the information carrier. The objective function for the image‐domain approach is designed to optimize the coherency of reflections in extended common‐image gathers. The function applies a penalty operator to the gathers, thus highlighting image inaccuracies arising from the velocity model error. Minimizing the objective function optimizes the model and improves the image quality. The gradient of the objective function is computed using the adjoint state method in a way similar to that in the analogous data‐domain implementation. We propose an image‐domain velocity‐model building method using extended common‐image‐point space‐ and time‐lag gathers constructed sparsely at reflections in the image. The gathers are effective in reconstructing the velocity model in complex geologic environments and can be used as an economical replacement for conventional common‐image gathers in wave‐equation tomography. A test on the Marmousi model illustrates successful updating of the velocity model using common‐image‐point gathers and resulting improved image quality.

Seismic envelope inversion: reduction of local minima and noise resistance

ABSTRACTWaveform inversion met severe challenge in retrieving long‐wavelength background structure. We have proposed to use envelope inversion to recover the large‐scale component of the model. Using the large‐scale background recovered by envelope inversion as new starting model, we can get much better result than the conventional full waveform inversion. By comparing the configurations of the misfit functional between the envelope inversion and the conventional waveform inversion, we show that envelope inversion can greatly reduce the local minimum problem. The combination of envelope inversion and waveform inversion can deliver more faithful and accurate final result with almost no extra computation cost compared to the conventional full waveform inversion. We also tested the noise resistance ability of envelope inversion to Gaussian noise and seismic interference noise. The results showed that envelope inversion is insensitive to Gaussian noise and, to a certain extent, insensitive to seismic interference noise. This indicates the robustness of this method and its potential use for noisy data.

Frequency-wavenumber domain phase inversion along reflection wavepaths

A background velocity model containing the correct low-wavenumber information is desired for both the quality of the migration image and the success of waveform inversion. To achieve this goal, the velocity is updated along the reflection wavepaths, rather than along both the reflection ellipses and transmission wavepaths as in conventional FWI. This method allows for reconstructing the low-wavenumber part of the background velocity model, even in the absence of long offsets and low-frequency component of the data. Moreover, in gradient-based iterative updates, instead of forming the data error conventionally, we propose to exploit the phase mismatch between the observed and the calculated data. The phase mismatch emphasizes a kinematic error and varies quasi-linearly with respect to the velocity error. The phase mismatch is computed (1) in the frequency–wavenumber (f–k) domain replacing the magnitudes of the calculated common shot gather by those of the observed one, and (2) in the temporal–spatial domain to form the difference between the transformed calculated common-shot gather and the observed one. The background velocity model inverted according to the proposed methods can serve as an improved initial velocity model for conventional waveform inversion. Tests with synthetic and field data show both the benefits and limitations of this method.

A new optimization approach for source-encoding full-waveform inversion

Waveform inversion is the method of choice for determining a highly heterogeneous subsurface structure. However, conventional waveform inversion requires that the wavefield for each source is computed separately. This makes it very expensive for realistic 3D seismic surveys. Source-encoding waveform inversion, in which the sources are modeled simultaneously, is considerably faster than conventional waveform inversion but suffers from artifacts. These artifacts can partly be removed by assigning random weights to the source wavefields. We found that the misfit function, and therefore also its gradient, for source-encoding waveform inversion is an unbiased random estimation of the misfit function used in conventional waveform inversion. We found a new method of source-encoding waveform inversion that takes into account the random nature of the gradients used in the optimization. In this new method, the gradient at each iteration is a weighted average of past gradients such that the most recent gradients have the largest weights with exponential decay. This way we damped the random fluctuations of the gradient by incorporating information from the previous iterations. We compared this new method with existing source-encoding waveform inversion methods as well as conventional waveform inversion and found that the model misfit reduction is faster and smoother than those of existing source-encoding waveform inversion methods, and it approaches the model misfit reduction obtained in conventional waveform inversion.

3D acoustic modelling and waveform inversion in the Laplace domain for an irregular sea floor using the Gaussian quadrature integration method

Abstract This paper explores the concept of 3D acoustic modelling and inversion in the Laplace domain. We used the Gaussian quadrature integration method to assemble the element mass matrix of a coarse regular rectangular element containing both water and subsurface parts, approximating the properties of these two parts without refining the element into smaller components. The purpose of this investigation was to model an irregular sea floor using 3D Laplace-domain modelling and inversion without increasing the computational cost incurred when using a small grid interval to precisely delineate the sea floor. The algorithm was verified by comparing the analytical solution to the modelled wavefield obtained using the conventional Laplace-domain modelling method and the modelled wavefield obtained using the Gaussian quadrature integration in a two-layer model (with 50-m, 100-m, and 200-m grid spacing and 5, 10, 20, and 40 Gaussian quadrature integration points) where an oblique water–subsurface interface is given. The results show that, under certain conditions, the Gaussian quadrature integration method can generate relatively accurate and consistent wavefields compared to the conventional modelling method. Moreover, we performed 3D Laplace-domain inversion with this Gaussian quadrature modelling technique to obtain synthetic data with a flat and irregular interface using a 100-m and 200-m grid. The results obtained under the conditions herein show that 3D Laplace-domain waveform inversion with Gaussian quadrature integration can generate a more accurate and cost-efficient velocity model compared to that obtained using conventional Laplace-domain waveform inversion.

Frequency-domain elastic full waveform inversion for VTI media

SUMMARY To describe subsurface structures in anisotropic media properly, particularly in transversely isotropic media with a vertical symmetry axis (VTI), which frequently appear in sedimentary basin environments, we develop a frequency-domain elastic full waveform inversion algorithm for 2-D VTI media. The inversion algorithm is based on the cell-based finite-difference modelling method and the adjoint state of the wave equation. Because the anisotropic inversion for VTI media deals with more elastic constants than the isotropic inversion, it is more prone to obtain local minimum solutions. For this reason, we may not succeed in properly describing the elastic constants of subsurface media if we only apply the standard inversion techniques to anisotropic waveform inversion. To compensate for the ill-posedness of the anisotropic waveform inversion, we couple elastic constants C11 and C33 based on Thomsen's relationship, which is also supported by the sensitivity analysis with respect to the parameters. To enhance the inversion results, we apply the frequency-selection strategy, moving from lower to higher frequencies and we carry out the inversion process over two stages. In both stages, all of the elastic constants are simultaneously optimized, as is done in the conventional waveform inversion. However, we only accept the inversion results for C11, C33 and C44 at the first stage, which will be used as the starting models for the second stage and C13 is reinitialized as a linearly increasing model. We apply our waveform inversion algorithm to a simple 3-layered model and a part of the overthrust model. For the 3-layered model, the first inversion stage is enough to yield reasonable inversion results for all of the elastic constants. For the overthrust model, the second stage is needed to enhance the inversion results for C13.

Delineating the rupture planes of an earthquake doublet using Source-Scanning Algorithm: application to the 2005 March 3 Ilan Doublet, northeast Taiwan

SUMMARY Correct identification of the fault plane(s) associated with an earthquake doublet is a very challenging problem because the pair of events often occurs in close space and time with almost the same magnitude. Most long-period waveforms of an earthquake doublet are severely tangled and thus unsuitable for conventional waveform inversion methods. In this study, we try to resolve this issue by utilizing the recently developed Source-Scanning Algorithm (SSA). The SSA systematically searches the model space for seismic sources whose times and locations are most compatible with the observed arrivals of large amplitudes on seismograms. The identification of a seismic source is based on the brightness function, which is defined as the summation of the normalized waveform amplitudes at the predicted arrival times at all stations. By illuminating the spatiotemporal distribution of asperities during an earthquake's source process, we are able to constrain the orientation of the rupture propagation that, in turn, leads to the identification of the fault plane. A series of synthetic experiments are performed to test SSA's resolution under various scenarios including different directions of rupture propagation, imperfect station coverage and short origin time difference between the two events of a doublet. Because only short-period records are needed in the analysis, the proposed method is best suited for an earthquake doublet with a short time gap between the two events. Using the 2005 Ilan doublet (the origin time difference is only 70 s) that occurred in northeast Taiwan as an example, we show that the trace of the brightest spots moves towards the west and infer the E–W-striking plane to be the actual fault plane.

2D acoustic‐elastic coupled waveform inversion in the Laplace domain

ABSTRACTAlthough waveform inversion has been intensively studied in an effort to properly delineate the Earth's structures since the early 1980s, most of the time‐ and frequency‐domain waveform inversion algorithms still have critical limitations in their applications to field data. This may be attributed to the highly non‐linear objective function and the unreliable low‐frequency components. To overcome the weaknesses of conventional waveform inversion algorithms, the acoustic Laplace‐domain waveform inversion has been proposed. The Laplace‐domain waveform inversion has been known to provide a long‐wavelength velocity model even for field data, which may be because it employs the zero‐frequency component of the damped wavefield and a well‐behaved logarithmic objective function. However, its applications have been confined to 2D acoustic media.We extend the Laplace‐domain waveform inversion algorithm to a 2D acoustic‐elastic coupled medium, which is encountered in marine exploration environments. In 2D acoustic‐elastic coupled media, the Laplace‐domain pressures behave differently from those of 2D acoustic media, although the overall features are similar to each other. The main differences are that the pressure wavefields for acoustic‐elastic coupled media show negative values even for simple geological structures unlike in acoustic media, when the Laplace damping constant is small and the water depth is shallow. The negative values may result from more complicated wave propagation in elastic media and at fluid‐solid interfaces.Our Laplace‐domain waveform inversion algorithm is also based on the finite‐element method and logarithmic wavefields. To compute gradient direction, we apply the back‐propagation technique. Under the assumption that density is fixed, P‐ and S‐wave velocity models are inverted from the pressure data. We applied our inversion algorithm to the SEG/EAGE salt model and the numerical results showed that the Laplace‐domain waveform inversion successfully recovers the long‐wavelength structures of the P‐ and S‐wave velocity models from the noise‐free data. The models inverted by the Laplace‐domain waveform inversion were able to be successfully used as initial models in the subsequent frequency‐domain waveform inversion, which is performed to describe the short‐wavelength structures of the true models.

Sequentially ordered single-frequency 2-D acoustic waveform inversion in the Laplace-Fourier domain

SUMMARY In the conventional frequency-domain waveform inversion, either multifrequency simultaneous inversion or sequential single-frequency inversion has been implemented. However, most conventional frequency-domain waveform inversion methods fail to recover background velocity when low-frequency information is missing. Recently, new waveform inversion techniques in the Laplace and Laplace–Fourier domain have been proposed to recover background velocity structure from data with insufficient low-frequency information. In such techniques, however, all frequencies are inverted simultaneously, and this requires large computational resources and long computation times. In this paper, we propose a sequentially ordered single-frequency 2-D acoustic waveform inversion using the logarithmic objective function in the Laplace–Fourier domain. Our algorithm sequentially inverts single-frequency data in the Laplace–Fourier domain, thus reducing computational resources. Unlike most conventional waveform inversion methods requiring an initial velocity model close to the true model, we propose a one-step waveform inversion method in seeking to find a final velocity structure from the simple initial model through a hybrid combination of the Laplace domain inversion and the Fourier domain inversion. We adopt and evaluate the multiloop algorithm by modifying the double-loop algorithm commonly used in the conventional frequency-domain waveform inversion. Using the multiloop algorithm repeating loop over frequencies, the quality of the inversion results can be improved and the decision problem of the number of iterations for each frequency can be overcome effectively. Because the sequential order of the Laplace–Fourier frequencies in a 2-D plane should be assigned for inverting Laplace–Fourier frequency data consecutively, we present three different sequential orders of Laplace–Fourier frequencies while considering the multiscale and layer-stripping approach, and we compare the inversion results from the numerical experiments. We applied the sequentially ordered single-frequency 2-D acoustic waveform inversion in the full Laplace–Fourier domain to the synthetic seismic data produced from complex structure model and field data. A realistic model could be recovered in an efficient and robust manner, even using the two-layer homogeneous velocity model as an initial model. The inverted velocity model from the field data was validated by examining the migrated image from the pre-stack depth migration and the flattening of the common-image gathers or by comparing the synthetic shot gather with the real shot gather. The proposed one-step waveform inversion algorithm can be easily extended to the sequential inversion of 3-D acoustic or elastic data in the full Laplace–Fourier domain.

Frequency-domain waveform inversion using an l1-norm objective function

In general, seismic waveform inversion adopts an objective function based on the l2-norm. However, waveform inversion using the l2-norm produces distorted results because the l2-norm is sensitive to statistically invalid data such as outliers. As an alternative, there have been several studies applying l1-norm-based objective functions to waveform inversion. Although waveform inversion based on the l1-norm is known to produce robust inversion results against specific outliers in the time domain, its effectiveness and characteristics are yet to be studied in the frequency domain. The present study proposes an algorithm for l1-norm-based waveform inversion in the frequency domain. The proposed algorithm employs a structure identical to those used in conventional frequency-domain waveform inversion algorithms that exploit the back-propagation technique, but displays robustness against outliers, which has been confirmed based on inversion of the synthetic Marmousi model. The characteristics and advantages of the l1-norm were analysed by comparing it with the l2-norm. In addition, inversion was performed on data containing outliers to examine the robustness against outliers. The effectiveness of removing outliers was verified by using the l1-norm to calculate the residual wavefield and its spectrum for the data containing outliers.

Comparison of scaling methods for waveform inversion

ABSTRACTWaveform inversion can lead to faint images for later times due to geometrical spreading. The proper scaling of the steepest‐descent direction can enhance faint images in waveform inversion results. We compare the effects of different scaling techniques in waveform inversion algorithms using the steepest‐descent method. For the scaling method we use the diagonal of the pseudo‐Hessian matrix, which can be applied in two different ways. One is to scale the steepest‐descent direction at each frequency independently. The other is to scale the steepest‐descent direction summed over the entire frequency band. The first method equalizes the steepest‐descent directions at different frequencies and minimizes the effects of the band‐limited source spectrum in waveform inversion. In the second method, since the steepest‐descent direction summed over the entire frequency band is divided by the diagonal of the pseudo‐Hessian matrix summed over the entire frequency band, the band‐limited property of the source wavelet spectrum still remains in the scaled steepest‐descent directions. The two scaling methods were applied to both standard and logarithmic waveform inversion. For standard waveform inversion, the method that scales the steepest‐descent direction at every frequency step gives better results than the second method. On the other hand, logarithmic waveform inversion is not sensitive to the scaling method, because taking the logarithm of wavefields automatically means that results for the steepest‐descent direction at each frequency are commensurate with each other. If once the steepest‐descent directions are equalized by taking the logarithm of wavefields in logarithmic waveform inversion, the additional equalizing effects by the scaling method are not as great as in conventional waveform inversion.

Waveform inversion using a logarithmic wavefield

Although waveform inversion has been studied extensively since its beginning [Formula: see text] ago, applications to seismic field data have been limited, and most of those applications have been for global-seismology- or engineering-seismology-scale problems, not for exploration-scale data. As an alternative to classical waveform inversion, we propose the use of a new, objective function constructed by taking the logarithm of wavefields, allowing consideration of three types of objective function, namely, amplitude only, phase only, or both. In our wave form inversion, we estimate the source signature as well as the velocity structure by including functions of amplitudes and phases of the source signature in the objective function. We compute the steepest-descent directions by using a matrix formalism derived from a frequency-domain, finite-element/finite-difference modeling technique. Our numerical algorithms are similar to those of reverse-time migration and waveform inversion based on the adjoint state of the wave equation. In order to demonstrate the practical applicability of our algorithm, we use a synthetic data set from the Marmousi model and seismic data collected from the Korean continental shelf. For noise-free synthetic data, the velocity structure produced by our inversion algorithm is closer to the true velocity structure than that obtained with conventional waveform inversion. When random noise is added, the inverted velocity model is also close to the true Marmousi model, but when frequencies below [Formula: see text] are removed from the data, the velocity structure is not as good as those for the noise-free and noisy data. For field data, we compare the time-domain synthetic seismograms generated for the velocity model inverted by our algorithm with real seismograms and find that the results show that our inversion algorithm reveals short-period features of the subsurface. Although we use wrapped phases in our examples, we still obtain reasonable results. We expect that if we were to use correctly unwrapped phases in the inversion algorithm, we would obtain better results.

Seismogram envelope inversion for the spatial distribution of high‐frequency energy radiation from the earthquake fault: Application to the 1994 far east off Sanriku earthquake, Japan

Incoherency of high‐frequency seismic waves leads to the idea to analyze seismogram envelopes disregarding phase information for the source study. We develop an inversion method for estimating high‐frequency (above 1 Hz) energy radiation from the earthquake fault and site amplification factors from observed mean square S‐wave seismogram envelopes, The inversion is executed by using the s‐wave envelope Green function from the onset to coda for a point shear dislocation source in a scattering medium, which is formulated based on an extended version of the radiative transfer theory. The use of the envelope Green function enables us to estimate the spatial distribution of high‐frequency energy radiation on a fault plane. This has been difficult by conventional waveform inversion methods. The rupture velocity and the duration time of energy radiation on each subfault are also estimated by grid search method. We apply this new method to the 1994 far east off Sanriku, northeastern Japan, earthquake (Mw 7.7), which ruptured a 90‐km segment of the plate boundary along the Japan Trench. Inverting observed seismogram envelopes at 10 strong‐motion stations in Japan, it is concluded that above 90% of the high‐frequency energy was radiated from the western half of the fault with the largest energy radiation near the deep‐side edge. The rupture velocity is estimated to be 2.7 km/s. The estimated site amplification factors, ranging between 0.3 and 15, are consistent with those independently estimated by the coda normalization method.