Chapter 6 Full waveform inversion and least-squares migration

A good starting model is imperative in full-waveform inversion (FWI) because it solves a least-squares inversion problem using a local gradient-based optimization method. A suboptimal starting model can result in cycle skipping leading to poor convergence and incorrect estimation of subsurface properties. This problem is especially crucial for salt models because the strong velocity contrasts create substantial time shifts in the modeled seismogram. Incorrect estimation of salt bodies leads to velocity inaccuracies in the sediments because the least-squares gradient aims to reduce traveltime differences without considering the sharp velocity jump between sediments and salt. We have developed a technique to estimate velocity models containing salt bodies using a combination of global and local optimization techniques. To stabilize the global optimization algorithm and keep it computationally tractable, we reduce the number of model parameters by using sparse parameterization formulations. The sparse formulation represents sediments using a set of interfaces and velocities across them, whereas a set of ellipses represents the salt body. We use very fast simulated annealing (VFSA) to minimize the misfit between the observed and synthetic data and estimate an optimal model in the sparsely parameterized space. The VFSA inverted model is then used as a starting model in FWI in which the sediments and salt body are updated in the least-squares sense. We partition model updates into sediment and salt updates in which the sediments are updated like conventional FWI, whereas the shape of the salt is updated by taking the zero crossing of an evolving level set surface. Our algorithm is tested on two 2D synthetic salt models, namely, the Sigsbee 2A model and a modified SEG Advanced Modeling Program (SEAM) Phase I model while fixing the top of the salt. We determine the efficiency of the VFSA inversion and imaging improvements from the level set FWI approach and evaluate a few sources of uncertainty in the estimation of salt shapes.

Download Full-text

The conjugate gradient method

The Leading Edge ◽

10.1190/tle37040296.1 ◽

2018 ◽

Vol 37 (4) ◽

pp. 296-298 ◽

Cited By ~ 1

Author(s):

Karl Schleicher

Keyword(s):

Least Squares ◽

Conjugate Gradient Method ◽

Conjugate Gradient ◽

Gradient Method ◽

Waveform Inversion ◽

Linear Inversion ◽

Large Computer ◽

Traveltime Tomography ◽

Full Waveform ◽

Least Squares Migration

The conjugate gradient method can be used to solve many large linear geophysical problems — for example, least-squares parabolic and hyperbolic Radon transform, traveltime tomography, least-squares migration, and full-waveform inversion (FWI) (e.g., Witte et al., 2018 ). This tutorial revisits the “Linear inversion tutorial” ( Hall, 2016 ) that estimated reflectivity by deconvolving a known wavelet from a seismic trace using least squares. This tutorial solves the same problem using the conjugate gradient method. This problem is easy to understand, and the concepts apply to other applications. The conjugate gradient method is often used to solve large problems because the least-squares algorithm is much more expensive — that is, even a large computer may not be able to find a useful solution in a reasonable amount of time.

Download Full-text

Illumination compensation of shadow zones in extended least squares migrated images by solving the linear inverse problem in tomographic full waveform inversion

10.1190/segam2019-3215094.1 ◽

2019 ◽

Author(s):

Rahul Sarkar ◽

Biondo Biondi

Keyword(s):

Inverse Problem ◽

Least Squares ◽

Waveform Inversion ◽

Full Waveform Inversion ◽

Illumination Compensation ◽

Linear Inverse Problem ◽

Full Waveform

Download Full-text

Land seismic multiparameter full waveform inversion in elastic VTI media by simultaneously interpreting body waves and surface waves with an optimal transport based objective function

Geophysical Journal International ◽

10.1093/gji/ggz414 ◽

2019 ◽

Vol 219 (3) ◽

pp. 1970-1988 ◽

Cited By ~ 3

Author(s):

Weiguang He ◽

Romain Brossier ◽

Ludovic Métivier ◽

René-Édouard Plessix

Keyword(s):

Objective Function ◽

Surface Waves ◽

Least Squares ◽

Optimal Transport ◽

Waveform Inversion ◽

Body Waves ◽

Model Parameters ◽

Full Waveform Inversion ◽

Initial Model ◽

Full Waveform

SUMMARY Land seismic multiparameter full waveform inversion in anisotropic media is challenging because of high medium contrasts and surface waves. With a data-residual least-squares objective function, the surface wave energy usually masks the body waves and the gradient of the objective function exhibits high values in the very shallow depths preventing from recovering the deeper part of the earth model parameters. The optimal transport objective function, coupled with a Gaussian time-windowing strategy, allows to overcome this issue by more focusing on phase shifts and by balancing the contributions of the different events in the adjoint-source and the gradients. We first illustrate the advantages of the optimal transport function with respect to the least-squares one, with two realistic examples. We then discuss a vertical transverse isotropic (VTI) example starting from a quasi 1-D isotropic initial model. Despite some cycle-skipping issues in the initial model, the inversion based on the windowed optimal transport approach converges. Both the near-surface complexities and the variations at depth are recovered.

Download Full-text

Microseismic Event Localization via Least-squares Full Waveform Inversion with Group Sparsity Constraints

79th EAGE Conference and Exhibition 2017 ◽

10.3997/2214-4609.201701261 ◽

2017 ◽

Cited By ~ 1

Author(s):

W. Gao ◽

M.D. Sacchi

Keyword(s):

Least Squares ◽

Waveform Inversion ◽

Full Waveform Inversion ◽

Group Sparsity ◽

Full Waveform ◽

Sparsity Constraints ◽

Microseismic Event ◽

Event Localization

Download Full-text

Application of optimal transport and the quadratic Wasserstein metric to full-waveform inversion

Geophysics ◽

10.1190/geo2016-0663.1 ◽

2018 ◽

Vol 83 (1) ◽

pp. R43-R62 ◽

Cited By ~ 59

Author(s):

Yunan Yang ◽

Björn Engquist ◽

Junzhe Sun ◽

Brittany F. Hamfeldt

Keyword(s):

Least Squares ◽

Optimal Transport ◽

Waveform Inversion ◽

Transportation Cost ◽

Velocity Model ◽

Seismic Inversion ◽

Full Waveform Inversion ◽

Wasserstein Metric ◽

Full Waveform ◽

Adjoint State

Conventional full-waveform inversion (FWI) using the least-squares norm as a misfit function is known to suffer from cycle-skipping issues that increase the risk of computing a local rather than the global minimum of the misfit. The quadratic Wasserstein metric has proven to have many ideal properties with regard to convexity and insensitivity to noise. When the observed and predicted seismic data are considered to be two density functions, the quadratic Wasserstein metric corresponds to the optimal cost of rearranging one density into the other, in which the transportation cost is quadratic in distance. Unlike the least-squares norm, the quadratic Wasserstein metric measures not only amplitude differences but also global phase shifts, which helps to avoid cycle-skipping issues. We have developed a new way of using the quadratic Wasserstein metric trace by trace in FWI and compare it with the global quadratic Wasserstein metric via the solution of the Monge-Ampère equation. We incorporate the quadratic Wasserstein metric technique into the framework of the adjoint-state method and apply it to several 2D examples. With the corresponding adjoint source, the velocity model can be updated using a quasi-Newton method. Numerical results indicate the effectiveness of the quadratic Wasserstein metric in alleviating cycle-skipping issues and sensitivity to noise. The mathematical theory and numerical examples demonstrate that the quadratic Wasserstein metric is a good candidate for a misfit function in seismic inversion.

Download Full-text

Full waveform inversion of marine reflection data in the plane‐wave domain

Geophysics ◽

10.1190/1.1444164 ◽

1997 ◽

Vol 62 (2) ◽

pp. 540-553 ◽

Cited By ~ 33

Author(s):

Susan E. Minkoff ◽

William W. Symes

Keyword(s):

Plane Wave ◽

Least Squares ◽

Waveform Inversion ◽

Viscoelastic Model ◽

Seismic Source ◽

P Wave ◽

Full Waveform Inversion ◽

Data Set ◽

Full Waveform ◽

Output Least Squares

Full waveform inversion of a p‐τ marine data set from the Gulf of Mexico provides estimates of the long‐wavelength P‐wave background velocity, anisotropic seismic source, and three high‐frequency elastic parameter reflectivities that explain 70% of the total seismic data and 90% of the data in an interval around the gas sand target. The forward simulator is based on a plane‐wave viscoelastic model for P‐wave propagation and primary reflections in a layered earth. Differential semblance optimization, a variant of output least‐squares inversion, successfully estimates the nonlinear P‐wave background velocity and linear reflectivities. Once an accurate velocity is estimated, output least‐squares inversion reestimates the reflectivities and an anisotropic seismic source simultaneously. The viscoelastic model predicts the amplitude‐versus‐angle trend in the data more accurately than does an elastic model. Simultaneous inversion for reflectivities and source explains substantially more of the actual data than does inversion for reflectivities with fixed source from an air‐gun modeler. The best reflectivity estimates conform to widely accepted lithologic relationships and closely match the filtered well logs.

Download Full-text