Nonlinear waveform inversion of plane‐wave seismograms in stratified elastic media

Geophysics ◽  
1991 ◽  
Vol 56 (5) ◽  
pp. 664-674 ◽  
Author(s):  
F. Kormendi ◽  
M. Dietrich
Keyword(s):  
Plane Wave ◽  
Least Squares ◽  
Wave Velocity ◽  
Generalized Least Squares ◽  
P Wave ◽  
Gradient Algorithm ◽  
Stratified Medium ◽  
Inversion Method ◽  
Inversion Algorithm ◽  
Frequency Wave

We present a method for determining the elastic parameters of a horizontally stratified medium from its plane‐wave reflectivity. The nonlinear inverse problem is iteratively solved by using a generalized least‐squares formalism. The proposed method uses the (relatively) fast convergence properties of the conjugate gradient algorithm and achieves computational efficiency through analytical solutions for calculating the reference and perturbational wavefields. The solution method is implemented in the frequency‐wave slowness domain and can be readily adapted to various source‐receiver configurations. The behavior of the algorithm conforms to the predictions of generalized least‐squares inverse theory: the inversion scheme yields satisfactory results as long as the correct velocity trends are introduced in the starting model. In practice, the inversion algorithm should be applied first in the precritical region because of the strong nonlinear behavior of postcritical data with respect to velocity perturbations. The suggested inversion strategy consists of first inverting for the density and P‐wave velocity (or P‐wave impedance) by considering plane waves in the low slowness region (near‐normal angles of incidence), then in optimizing for the S‐wave velocity by progressively including contributions from the high slowness region (steep angles of incidence). Numerical experiments performed with noise‐free synthetic data prove that the proposed inversion method satisfactorialy reconstructs the elastic properties of a stratified medium from a limited set of plane‐wave components, at a reasonable computing cost.

Frequency‐wavenumber elastic inversion of marine seismic data

Geophysics ◽  
1994 ◽  
Vol 59 (12) ◽  
pp. 1868-1881 ◽  
Author(s):  
Huasheng Zhao ◽  
Bjørn Ursin ◽  
Lasse Amundsen
Keyword(s):  
Wave Velocity ◽  
Reference Data ◽  
Jacobian Matrix ◽  
Synthetic Data ◽  
Velocity Model ◽  
P Wave ◽  
Stratified Medium ◽  
Inversion Method ◽  
S Wave ◽  
Time Space

We present an inversion method for determining the velocities, densities, and layer thicknesses of a horizontally stratified medium with an acoustic layer at the top and a stack of elastic layers below. The multioffset reflection response of the medium generated by a compressional point source is transformed from the time‐space domain into the frequency‐wavenumber domain where the inversion is performed by minimizing the difference between the reference data and the modeled data using a least‐squares technique. The forward modeling is based on the reflectivity method where the solution for each frequency‐wavenumber component is found by computing the generalized reflection and transmission matrices recursively. The gradient of the objective function is computed from analytical expressions of the Jacobian matrix derived directly from the recursive modeling equations. The partial derivatives of the reflection response of the stratified medium are then computed simultaneously with the reflection response by layer‐recursive formulas. The limited‐aperture and discretization effects in time and space of the reference data are included by applying a pair of frequency and wavenumber dependent filters to the predicted data and to the Jacobian matrix at each iteration. Numerical experiments performed with noise‐free synthetic data prove that the proposed inversion method satisfactorily reconstructs the elastic parameters of a stratified medium. The low‐frequency trends of the S‐wave velocity and density are found when the initial P‐wave velocity model gives approximately correct traveltimes. The convergence of the iterative minimization algorithm is fast.

Bayesian linearized AVO inversion

Geophysics ◽  
2003 ◽  
Vol 68 (1) ◽  
pp. 185-198 ◽  
Author(s):  
Arild Buland ◽  
Henning Omre
Keyword(s):  
Wave Velocity ◽  
Posterior Distribution ◽  
Acoustic Impedance ◽  
Velocity Ratio ◽  
P Wave ◽  
Inversion Method ◽  
Inversion Algorithm ◽  
S Wave ◽  
Data Set ◽  
Avo Inversion

A new linearized AVO inversion technique is developed in a Bayesian framework. The objective is to obtain posterior distributions for P‐wave velocity, S‐wave velocity, and density. Distributions for other elastic parameters can also be assessed—for example, acoustic impedance, shear impedance, and P‐wave to S‐wave velocity ratio. The inversion algorithm is based on the convolutional model and a linearized weak contrast approximation of the Zoeppritz equation. The solution is represented by a Gaussian posterior distribution with explicit expressions for the posterior expectation and covariance; hence, exact prediction intervals for the inverted parameters can be computed under the specified model. The explicit analytical form of the posterior distribution provides a computationally fast inversion method. Tests on synthetic data show that all inverted parameters were almost perfectly retrieved when the noise approached zero. With realistic noise levels, acoustic impedance was the best determined parameter, while the inversion provided practically no information about the density. The inversion algorithm has also been tested on a real 3‐D data set from the Sleipner field. The results show good agreement with well logs, but the uncertainty is high.

This issue of GEOPHYSICS

Geophysics ◽  
1996 ◽  
Vol 61 (5) ◽  
pp. 1245-1246
Keyword(s):  
Least Squares ◽  
Wave Velocity ◽  
Inverse Modeling ◽  
Transversely Isotropic ◽  
P Wave ◽  
Elastic Parameters ◽  
P Wave Velocity ◽  
Iterative Inversion ◽  
Best Fitting ◽  
Modeling Material

Okoye et al. develop a least-squares iterative inversion technique determining of the elastic parameters δ* and vertical P-wave velocity (α0) of any transversely isotropic modeling material in the laboratory. The anisotropic inverse modeling technique finds the best fitting solution and implements analytical rather than numerical differentiations to optimize the accuracy of the results.

scholarly journals Travel-Time Inversion Method of Converted Shear Waves Using RayInvr Algorithm

2021 ◽  
Vol 11 (8) ◽  
pp. 3571
Author(s):  
Genggeng Wen ◽  
Kuiyuan Wan ◽  
Shaohong Xia ◽  
Huilong Xu ◽  
Chaoyan Fan ◽  
...  
Keyword(s):  
Travel Time ◽  
Wave Velocity ◽  
P Wave ◽  
Inversion Method ◽  
Time Inversion ◽  
Ocean Bottom Seismometer ◽  
S Wave ◽  
S Waves ◽  
Inversion Model ◽  
Travel Time Inversion

The detailed studies of converted S-waves recorded on the Ocean Bottom Seismometer (OBS) can provide evidence for constraining lithology and geophysical properties. However, the research of converted S-waves remains a weakness, especially the S-waves’ inversion. In this study, we applied a travel-time inversion method of converted S-waves to obtain the crustal S-wave velocity along the profile NS5. The velocities of the crust are determined by the following four aspects: (1) modelling the P-wave velocity, (2) constrained sediments Vp/Vs ratios and S-wave velocity using PPS phases, (3) the correction of PSS phases’ travel-time, and (4) appropriate parameters and initial model are selected for inversion. Our results show that the vs. and Vp/Vs of the crust are 3.0–4.4 km/s and 1.71–1.80, respectively. The inversion model has a similar trend in velocity and Vp/Vs ratios with the forward model, due to a small difference with ∆Vs of 0.1 km/s and ∆Vp/Vs of 0.03 between two models. In addition, the high-resolution inversion model has revealed many details of the crustal structures, including magma conduits, which further supports our method as feasible.

Simultaneous inversion of prestack seismic data for rock properties using simulated annealing

Geophysics ◽  
2002 ◽  
Vol 67 (6) ◽  
pp. 1877-1885 ◽  
Author(s):  
Xin‐Quan Ma
Keyword(s):  
Simulated Annealing ◽  
Seismic Data ◽  
Low Frequency ◽  
Optimization Procedure ◽  
P Wave ◽  
Rock Properties ◽  
Inversion Method ◽  
Inversion Algorithm ◽  
Simultaneous Inversion ◽  
Reflection Seismic

A new prestack inversion algorithm has been developed to simultaneously estimate acoustic and shear impedances from P‐wave reflection seismic data. The algorithm uses a global optimization procedure in the form of simulated annealing. The goal of optimization is to find a global minimum of the objective function, which includes the misfit between synthetic and observed prestack seismic data. During the iterative inversion process, the acoustic and shear impedance models are randomly perturbed, and the synthetic seismic data are calculated and compared with the observed seismic data. To increase stability, constraints have been built into the inversion algorithm, using the low‐frequency impedance and background Vs/Vp models. The inversion method has been successfully applied to synthetic and field data examples to produce acoustic and shear impedances comparable to log data of similar bandwidth. The estimated acoustic and shear impedances can be combined to derive other elastic parameters, which may be used for identifying of lithology and fluid content of reservoirs.

scholarly journals High-resolution prestack seismic inversion using a hybrid FISTA least-squares strategy

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. R185-R195 ◽  
Author(s):  
Daniel O. Pérez ◽  
Danilo R. Velis ◽  
Mauricio D. Sacchi
Keyword(s):  
High Resolution ◽  
Least Squares ◽  
Large Scale ◽  
Cost Effective ◽  
Seismic Inversion ◽  
Iterative Solvers ◽  
Gradient Algorithm ◽  
Inversion Method ◽  
Thresholding Algorithm ◽  
Sparse Solutions

A new inversion method to estimate high-resolution amplitude-versus-angle attributes (AVA) attributes such as intercept and gradient from prestack data is presented. The proposed technique promotes sparse-spike reflectivities that, when convolved with the source wavelet, fit the observed data. The inversion is carried out using a hybrid two-step strategy that combines fast iterative shrinkage-thresholding algorithm (FISTA) and a standard least-squares (LS) inversion. FISTA, which can be viewed as an extension of the classical gradient algorithm, provides sparse solutions by minimizing the misfit between the modeled and the observed data, and the [Formula: see text]-norm of the solution. FISTA is used to estimate the location in time of the main reflectors. Then, LS is used to retrieve the appropriate reflectivity amplitudes that honor the data. FISTA, like other iterative solvers for [Formula: see text]-norm regularization, does not require matrices in explicit form, making it easy to apply, economic in computational terms, and adequate for solving large-scale problems. As a consequence, the FISTA+LS strategy represents a simple and cost-effective new procedure to solve the AVA inversion problem. Results on synthetic and field data show that the proposed hybrid method can obtain high-resolution AVA attributes from noisy observations, making it an interesting alternative to conventional methods.

Direct nonlinear inversion of multiparameter 1D elastic media using the inverse scattering series

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCD15-WCD27 ◽  
Author(s):  
Haiyan Zhang ◽  
Arthur B. Weglein
Keyword(s):  
Wave Velocity ◽  
Material Properties ◽  
Value Added ◽  
P Wave ◽  
Added Value ◽  
Inversion Method ◽  
Elastic Media ◽  
Nonlinear Inversion ◽  
Model Matching ◽  
S Wave

In the direct nonlinear inversion method and in algorithms for 1D elastic media, P-wave velocity, S-wave velocity, and density are depth dependent. “Direct nonlinear” means that the method uses explicit formulas that (1) input data and directly output changes in material properties without the need for indirect procedures such as model matching, searching, optimization, or other assumed aligned objectives or proxies and that (2) the algorithms recognize and directly invert the intrinsic nonlinear relationship between changes in material properties and the recorded reflection wavefields. To achieve full elastic inversion, all components of data (such as PP, SP, and SS data) are needed. The method assumes that only data and reference medium propertiesare input, and terms in the inverse series for moving mislocated reflectors resulting from the linear inverse term are separated from amplitude correction terms. Although in principle this direct inversion approach requires all components of elastic data, synthetic tests indicate that a consistent value-added result may be achieved given only PP data measurements, as long as the PP data are used to approximately synthesize the PS and SP components. Further value would be derived from measuring all components of the data as the method requires. If all components of data are available, one consistent method can solve for all of the second terms (the first terms beyond linear). The explicit nonlinear inversion formulas provide an unambiguous data requirement message as well as conceptual and practical added value beyond both linear approaches and all indirect methods.

Inversion of the shear velocity of the cement in cased borehole through ultrasonic flexural waves

Geophysics ◽  
2017 ◽  
Vol 82 (2) ◽  
pp. D57-D68 ◽  
Author(s):  
Feilong Xu ◽  
Hengshan Hu
Keyword(s):  
Wave Velocity ◽  
Error Rate ◽  
Shear Velocity ◽  
P Wave ◽  
Inversion Method ◽  
Flexural Waves ◽  
S Wave ◽  
Additional Time ◽  
Full Waveforms ◽  
Cased Borehole

Young’s elastic modulus and Poisson’s ratio of the cement in a cased borehole are important in the prediction of the cement sheath integrity under hydraulic fracturing. Although these mechanical properties can be derived in principle from the bulk velocities, the inversion of these velocities of the cement from the received full waveforms remains a challenging problem, especially the S-wave velocity. We have developed an inversion method based on the round-trip traveltimes of the leaked flexural waves (TTL) to invert the bulk velocities of the medium behind the casing. The traveltime difference between the casing wave and the delayed casing wave is the additional time for the leaked wave to travel in the interlayer to the formation and back to the casing. To demonstrate the effectiveness of this method, synthetic full waveforms with a changing interlayer are calculated when an ultrasonic acoustic beam is incident obliquely on the casing. The traveltimes of the wave packets are picked from the envelope curve of the full waveform and then used to invert the bulk velocities in the TTL method. The inverted S-wave velocity of cement is quite accurate with an error rate smaller than 3%, no matter whether the cement is of the ordinary, heavy, or light type. When the interlayer is mud, the P-wave is inverted with an error rate of less than 2%. The P-wave velocity is inverted roughly with an error rate of approximately 10% when the medium behind the casing is light cement.

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

Geophysics ◽  
1997 ◽  
Vol 62 (2) ◽  
pp. 540-553 ◽  
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.

Absorbing boundary conditions for elastic waves

Geophysics ◽  
1991 ◽  
Vol 56 (2) ◽  
pp. 231-241 ◽  
Author(s):  
R. L. Higdon
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Wave Velocity ◽  
Elastic Waves ◽  
Numerical Models ◽  
Absorbing Boundary Conditions ◽  
P Wave ◽  
Stratified Medium ◽  
Computational Domain ◽  
Absorbing Boundary

Absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains. The boundary conditions developed here for elastic waves are generalizations of ones developed earlier for acoustic waves. These conditions are based on compositions of simple first‐order differential operators. The formulas can be applied without modification to problems in both two and three dimensions. The boundary conditions are stable for all values of the ratio of P‐wave velocity to S‐wave velocity, and they are effective near a free surface and in a horizontally stratified medium. The boundary conditions are approximated with simple finite‐difference equations that use values of the solution only along grid lines perpendicular to the boundary. This property facilitates implementation, especially near a free surface and at other corners of the computational domain.

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