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1. Liouville partial-differential-equation methods for computing 2D complex multivalued eikonals in attenuating media

   https://doi.org/10.1190/GEO2021-0113.1  

   Leung, Shingyu ; Hu, Jiangtao ; Qian, Jianliang ( March 2022 , GEOPHYSICS)

   We have developed a Liouville partial-differential-equation (PDE)-based method for computing complex-valued eikonals in real phase space in the multivalued sense in attenuating media with frequency-independent qualify factors, where the new method computes the real and imaginary parts of the complex-valued eikonal in two steps by solving Liouville equations in real phase space. Because the earth is composed of attenuating materials, seismic waves usually attenuate so that seismic data processing calls for properly treating the resulting energy losses and phase distortions of wave propagation. In the regime of high-frequency asymptotics, the complex-valued eikonal is one essential ingredient for describing wave propagation in attenuating media because this unique quantity summarizes two wave properties into one function: Its real part describes the wave kinematics and its imaginary part captures the effects of phase dispersion and amplitude attenuation. Because some popular ordinary-differential-equation (ODE)-based ray-tracing methods for computing complex-valued eikonals in real space distribute the eikonal function irregularly in real space, we are motivated to develop PDE-based Eulerian methods for computing such complex-valued eikonals in real space on regular meshes. Therefore, we solved novel paraxial Liouville PDEs in real phase space so that we can compute the real and imaginary parts of the complex-valued eikonal in the multivalued sense on regular meshes. We call the resulting method the Liouville PDE method for complex-valued multivalued eikonals in attenuating media; moreover, this new method provides a unified framework for Eulerianizing several popular approximate real-space ray-tracing methods for complex-valued eikonals, such as viscoacoustic ray tracing, real viscoelastic ray tracing, and real elastic ray tracing. In addition, we also provide Liouville PDE formulations for computing multivalued ray amplitudes in a weakly viscoacoustic medium. Numerical examples, including a synthetic gas-cloud model, demonstrate that our methods yield highly accurate complex-valued eikonals in the multivalued sense. 

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2. Ray-illumination compensation for adjoint-state first-arrival traveltime tomography

   https://doi.org/10.1190/GEO2020-0140.1  

   Hu, Jiangtao ; Qian, Jianliang ; Cao, Junxing ; Wang, Xingjian ; Wang, Huazhong ; Leung, Shingyu ( September 2021 , GEOPHYSICS)

   First-arrival traveltime tomography is an essential method for obtaining near-surface velocity models. The adjoint-state first-arrival traveltime tomography is appealing due to its straightforward implementation, low computational cost, and low memory consumption. Because solving the point-source isotropic eikonal equation by either ray tracers or eikonal solvers intrinsically corresponds to emanating discrete rays from the source point, the resulting traveltime gradient is singular at the source point, and we denote such a singular pattern the imprint of ray-illumination. Because the adjoint-state equation propagates traveltime residuals back to the source point according to the negative traveltime gradient, the resulting adjoint state will inherit such an imprint of ray-illumination, leading to singular gradient-descent directions when updating the velocity model in the adjoint-state traveltime tomography. To mitigate this imprint, we solve the adjoint-state equation twice but with different boundary conditions: one being taken to be regular data residuals and the other taken to be ones uniformly, so that we are able to use the latter adjoint state to normalize the regular adjoint state and we further use the normalized quantity to serve as the gradient direction to update the velocity model; we call this process ray-illumination compensation. To overcome the issue of limited aperture, we have developed a spatially varying regularization method to stabilize the new gradient direction. A synthetic example demonstrates that our method is able to mitigate the imprint of ray-illumination, remove the footprint effect near source points, and provide uniform velocity updates along raypaths. A complex example extracted from the Marmousi2 model and a migration example illustrate that the new method accurately recovers the velocity model and that an offset-dependent inversion strategy can further improve the quality of recovered velocity models. 

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3. Accelerating full waveform inversion via source stacking and cross-correlations

   https://doi.org/10.1093/gji/ggz437  

   Romanowicz, Barbara ; Chen, Li-Wei ; French, Scott W. ( October 2019 , Geophysical Journal International)

   Accurate synthetic seismic wavefields can now be computed in 3-D earth models using the spectral element method (SEM), which helps improve resolution in full waveform global tomography. However, computational costs are still a challenge. These costs can be reduced by implementing a source stacking method, in which multiple earthquake sources are simultaneously triggered in only one teleseismic SEM simulation. One drawback of this approach is the perceived loss of resolution at depth, in particular because high-amplitude fundamental mode surface waves dominate the summed waveforms, without the possibility of windowing and weighting as in conventional waveform tomography.

   This can be addressed by redefining the cost-function and computing the cross-correlation wavefield between pairs of stations before each inversion iteration. While the Green’s function between the two stations is not reconstructed as well as in the case of ambient noise tomography, where sources are distributed more uniformly around the globe, this is not a drawback, since the same processing is applied to the 3-D synthetics and to the data, and the source parameters are known to a good approximation. By doing so, we can separate time windows with large energy arrivals corresponding to fundamental mode surface waves. This opens the possibility of designing a weighting scheme to bring out the contribution of overtones and body waves. It also makes it possible to balance the contributions of frequently sampled paths versus rarely sampled ones, as in more conventional tomography.

   Here we present the results of proof of concept testing of such an approach for a synthetic 3-component long period waveform data set (periods longer than 60 s), computed for 273 globally distributed events in a simple toy 3-D radially anisotropic upper mantle model which contains shear wave anomalies at different scales. We compare the results of inversion of 10 000 s long stacked time-series, starting from a 1-D model, using source stacked waveforms and station-pair cross-correlations of these stacked waveforms in the definition of the cost function. We compute the gradient and the Hessian using normal mode perturbation theory, which avoids the problem of cross-talk encountered when forming the gradient using an adjoint approach. We perform inversions with and without realistic noise added and show that the model can be recovered equally well using one or the other cost function.

   The proposed approach is computationally very efficient. While application to more realistic synthetic data sets is beyond the scope of this paper, as well as to real data, since that requires additional steps to account for such issues as missing data, we illustrate how this methodology can help inform first order questions such as model resolution in the presence of noise, and trade-offs between different physical parameters (anisotropy, attenuation, crustal structure, etc.) that would be computationally very costly to address adequately, when using conventional full waveform tomography based on single-event wavefield computations.

    

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4. Embedding properties of network realizations of dissipative reduced order models

   Druskin, Guddati ( July 2022 , HOUSEHOLDER SYMPOSIUM XXI)

   Embedding properties of network realizations of dissipative reduced order models Jörn Zimmerling, Mikhail Zaslavsky,Rob Remis, Shasri Moskow, Alexander Mamonov, Murthy Guddati, Vladimir Druskin, and Liliana Borcea Mathematical Sciences Department, Worcester Polytechnic Institute https://www.wpi.edu/people/vdruskin Abstract Realizations of reduced order models of passive SISO or MIMO LTI problems can be transformed to tridiagonal and block-tridiagonal forms, respectively, via dierent modications of the Lanczos algorithm. Generally, such realizations can be interpreted as ladder resistor-capacitor-inductor (RCL) networks. They gave rise to network syntheses in the rst half of the 20th century that was at the base of modern electronics design and consecutively to MOR that tremendously impacted many areas of engineering (electrical, mechanical, aerospace, etc.) by enabling ecient compression of the underlining dynamical systems. In his seminal 1950s works Krein realized that in addition to their compressing properties, network realizations can be used to embed the data back into the state space of the underlying continuum problems. In more recent works of the authors Krein's ideas gave rise to so-called nite-dierence Gaussian quadrature rules (FDGQR), allowing to approximately map the ROM state-space representation to its full order continuum counterpart on a judicially chosen grid. Thus, the state variables can be accessed directly from the transfer function without solving the full problem and even explicit knowledge of the PDE coecients in the interior, i.e., the FDGQR directly learns" the problem from its transfer function. This embedding property found applications in PDE solvers, inverse problems and unsupervised machine learning. Here we show a generalization of this approach to dissipative PDE problems, e.g., electromagnetic and acoustic wave propagation in lossy dispersive media. Potential applications include solution of inverse scattering problems in dispersive media, such as seismic exploration, radars and sonars. To x the idea, we consider a passive irreducible SISO ROM fn(s) = Xn j=1 yi s + σj , (62) assuming that all complex terms in (62) come in conjugate pairs. We will seek ladder realization of (62) as rjuj + vj − vj−1 = −shˆjuj , uj+1 − uj + ˆrj vj = −shj vj , (63) for j = 0, . . . , n with boundary conditions un+1 = 0, v1 = −1, and 4n real parameters hi, hˆi, ri and rˆi, i = 1, . . . , n, that can be considered, respectively, as the equivalent discrete inductances, capacitors and also primary and dual conductors. Alternatively, they can be viewed as respectively masses, spring stiness, primary and dual dampers of a mechanical string. Reordering variables would bring (63) into tridiagonal form, so from the spectral measure given by (62 ) the coecients of (63) can be obtained via a non-symmetric Lanczos algorithm written in J-symmetric form and fn(s) can be equivalently computed as fn(s) = u1. The cases considered in the original FDGQR correspond to either (i) real y, θ or (ii) real y and imaginary θ. Both cases are covered by the Stieltjes theorem, that yields in case (i) real positive h, hˆ and trivial r, rˆ, and in case (ii) real positive h,r and trivial hˆ,rˆ. This result allowed us a simple interpretation of (62) as the staggered nite-dierence approximation of the underlying PDE problem [2]. For PDEs in more than one variables (including topologically rich data-manifolds), a nite-dierence interpretation is obtained via a MIMO extensions in block form, e.g., [4, 3]. The main diculty of extending this approach to general passive problems is that the Stieltjes theory is no longer applicable. Moreover, the tridiagonal realization of a passive ROM transfer function (62) via the ladder network (63) cannot always be obtained in port-Hamiltonian form, i.e., the equivalent primary and dual conductors may change sign [1]. 100 Embedding of the Stieltjes problems, e.g., the case (i) was done by mapping h and hˆ into values of acoustic (or electromagnetic) impedance at grid cells, that required a special coordinate stretching (known as travel time coordinate transform) for continuous problems. Likewise, to circumvent possible non-positivity of conductors for the non-Stieltjes case, we introduce an additional complex s-dependent coordinate stretching, vanishing as s → ∞ [1]. This stretching applied in the discrete setting induces a diagonal factorization, removes oscillating coecients, and leads to an accurate embedding for moderate variations of the coecients of the continuum problems, i.e., it maps discrete coecients onto the values of their continuum counterparts. Not only does this embedding yields an approximate linear algebraic algorithm for the solution of the inverse problems for dissipative PDEs, it also leads to new insight into the properties of their ROM realizations. We will also discuss another approach to embedding, based on Krein-Nudelman theory [5], that results in special data-driven adaptive grids. References [1] Borcea, Liliana and Druskin, Vladimir and Zimmerling, Jörn, A reduced order model approach to inverse scattering in lossy layered media, Journal of Scientic Computing, V. 89, N1, pp. 136,2021 [2] Druskin, Vladimir and Knizhnerman, Leonid, Gaussian spectral rules for the three-point second dierences: I. A two-point positive denite problem in a semi-innite domain, SIAM Journal on Numerical Analysis, V. 37, N 2, pp.403422, 1999 [3] Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Distance preserving model order reduction of graph-Laplacians and cluster analysis, Druskin, Vladimir and Mamonov, Alexander V and Zaslavsky, Mikhail, Journal of Scientic Computing, V. 90, N 1, pp 130, 2022 [4] Druskin, Vladimir and Moskow, Shari and Zaslavsky, Mikhail LippmannSchwingerLanczos algorithm for inverse scattering problems, Inverse Problems, V. 37, N. 7, 2021, [5] Mark Adolfovich Nudelman The Krein String and Characteristic Functions of Maximal Dissipative Operators, Journal of Mathematical Sciences, 2004, V 124, pp 49184934 Go back to Plenary Speakers Go back to Speakers Go back 

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5. Double-difference seismic attenuation tomography method and its application to The Geysers geothermal field, California

   https://doi.org/10.1093/gji/ggab017  

   Guo, Hao ; Thurber, Clifford ( February 2021 , Geophysical Journal International)

   SUMMARY Knowledge of attenuation structure is important for understanding subsurface material properties. We have developed a double-difference seismic attenuation (DDQ) tomography method for high-resolution imaging of 3-D attenuation structure. Our method includes two main elements, the inversion of event-pair differential ${t^*}$ ($d{t^*}$) data and 3-D attenuation tomography with the $d{t^*}$ data. We developed a new spectral ratio method that jointly inverts spectral ratio data from pairs of events observed at a common set of stations to determine the $d{t^*}$ data. The spectral ratio method cancels out instrument and site response terms, resulting in more accurate $d{t^*}$ data compared to absolute ${t^*}$ from traditional methods using individual spectra. Synthetic tests show that the inversion of $d{t^*}$ data using our spectral ratio method is robust to the choice of source model and a moderate degree of noise. We modified an existing velocity tomography code so that it can invert $d{t^*}$ data for 3-D attenuation structure. We applied the new method to The Geyser geothermal field, California, which has vapour-dominated reservoirs and a long history of water injection. A new Qp model at The Geysers is determined using P-wave data of earthquakes in 2011, using our updated earthquake locations and Vp model. By taking advantage of more accurate $d{t^*}$ data and the cancellation of model uncertainties along the common paths outside of the source region, the DDQ tomography method achieves higher resolution, especially in the earthquake source regions, compared to the standard tomography method using ${t^*}$ data. This is validated by both the real and synthetic data tests. Our Qp and Vp models show consistent variations in a normal temperature reservoir that can be explained by variations in fracturing, permeability and fluid saturation and/or steam pressure. A prominent low-Qp and Vp zone associated with very active seismicity is imaged within a high temperature reservoir at depths below 2 km. This anomalous zone is likely partially saturated with injected fluids. 

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