Seismic waveform inversion in the frequency domain, Part 1: Theory and verification in a physical scale model

Geophysics ◽  
1999 ◽  
Vol 64 (3) ◽  
pp. 888-901 ◽  
Author(s):  
R. Gerhard Pratt
Keyword(s):  
Finite Difference ◽  
Frequency Domain ◽  
Waveform Inversion ◽  
Forward Modeling ◽  
Scale Model ◽  
Model Data ◽  
Seismic Waveform ◽  
Physical Scale ◽  
Wavefield Inversion ◽  
Selection Of

Seismic waveforms contain much information that is ignored under standard processing schemes; seismic waveform inversion seeks to use the full information content of the recorded wavefield. In this paper I present, apply, and evaluate a frequency‐space domain approach to waveform inversion. The method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data. The model data are computed using a full‐wave equation, viscoacoustic, frequency‐domain, finite‐difference method. Ray asymptotics are avoided, and higher‐order effects such as diffractions and multiple scattering are accounted for automatically. The theory of frequency‐domain waveform/wavefield inversion can be expressed compactly using a matrix formalism that uses finite‐difference/finite‐element frequency‐domain modeling equations. Expressions for fast, local descent inversion using back‐propagation techniques then follow naturally. Implementation of these methods depends on efficient frequency‐domain forward‐modeling solutions; these are provided by recent developments in numerical forward modeling. The inversion approach resembles prestack, reverse‐time migration but differs in that the problem is formulated in terms of velocity (not reflectivity), and the method is fully iterative. I illustrate the practical application of the frequency‐domain waveform inversion approach using tomographic seismic data from a physical scale model. This allows a full evaluation and verification of the method; results with field data are presented in an accompanying paper. Several critical processes contribute to the success of the method: the estimation of a source signature, the matching of amplitudes between real and synthetic data, the selection of a time window, and the selection of suitable sequence of frequencies in the inversion. An initial model for the inversion of the scale model data is provided using standard traveltime tomographic methods, which provide a robust but low‐resolution image. Twenty‐five iterations of wavefield inversion are applied, using five discrete frequencies at each iteration, moving from low to high frequencies. The final results exhibit the features of the true model at subwavelength scale and account for many of the details of the observed arrivals in the data.

Seismic waveform inversion in the frequency domain, Part 2: Fault delineation in sediments using crosshole data

Geophysics ◽  
1999 ◽  
Vol 64 (3) ◽  
pp. 902-914 ◽  
Author(s):  
R. Gerhard Pratt ◽  
Richard M. Shipp
Keyword(s):  
Frequency Domain ◽  
Field Data ◽  
Sedimentary Environment ◽  
Waveform Inversion ◽  
Normal Fault ◽  
Velocity Model ◽  
Viscous Effects ◽  
Seismic Waveform ◽  
Wavefield Inversion ◽  
High Degree

A crosshole experiment was carried out in a layered sedimentary environment in which a normal fault is known to cut through the section. Initial traveltime inversions produced stable but low‐resolution images from which the fault could be only vaguely inferred. To image the fault, wavefield inversion was used to produce a velocity model consistent with the detailed phase and amplitude of the data at a number of frequencies. Our wavefield inversion scheme uses a classical, descent‐type algorithm for decreasing the data misfit by iteratively computing the gradient of this misfit by repeated forward and backward propagations. Our propagator is a full‐wave equation, frequency‐domain, acoustic, finite‐difference method. The use of the frequency‐space domain yields computational advantages for multisource data and allows an easy incorporation of viscous effects. By running wavefield inversion on the field data, a quantitative velocity image was produced that yielded a significantly improved image of the fault (when compared with the original traveltime inversions). Because the original field data were noisy and contained a high degree of multiple scattering (from the layering of the sediments), the transmitted arrivals were selectively windowed to enhance the image. The sediments at the site were strongly attenuating; we therefore used a viscoacoustic model during the modeling and the inversion that correctly simulated the observed decrease in amplitude with increasing frequency and source‐receiver offset. Furthermore, since the traveltime inversion indicated a high degree of anisotropy at the site, a fixed, homogeneous level of anisotropy was used during the inversion. Tests at varying levels of anisotropy confirmed the improvement in image quality and in data fit when anisotropy was incorporated. The final image was verified by examining the distribution of the residuals in the frequency domain, by comparing time‐domain modeled wavefields with the observed data, and by direct comparison with borehole logs.

Use of Scale-Model Aids in Solution of Complex Air-Flow Design Problem

1965 ◽  
Author(s):  
Arthur Oppenheim ◽  
Frederic M. Oran
Keyword(s):  
Scale Construction ◽  
Scale Model ◽  
Inlet Section ◽  
Engine Test ◽  
Model Data ◽  
Jet Engine ◽  
Basic Fluid ◽  
Air Stream ◽  
Step Procedure ◽  
Selection Of

This paper describes a method used to design the inlet section of a jet-engine test cell. Calculations, accomplished with the use of standard, tabulated flow formulas and coefficients which can be found in basic fluid-flow texts, are shown. The step-by-step procedure enables the reader to use the methods and formulas in this paper as a guide for solving similar problems. Selection of an optimum turning-vane configuration is described herein by the use of tufts of cotton in the air stream of the model. A table is included which compares the model data with actual full-scale construction at various stages.

3-D physical modeling and pseudospectral simulation of seismic common‐source data volumes

Geophysics ◽  
1993 ◽  
Vol 58 (1) ◽  
pp. 121-133 ◽  
Author(s):  
How‐Wei Chen ◽  
George A. McMechan
Keyword(s):  
Three Dimensional ◽  
Scale Model ◽  
Adjustment Process ◽  
Common Source ◽  
Model Data ◽  
Iterative Model ◽  
Physical Scale ◽  
Out Of Plane ◽  
Potential Applicability ◽  
Model Adjustment

Numerical simulation of common‐source seismic responses of arbitrarily complicated three‐dimensional (3-D) variable‐velocity structures is implemented by pseudospectral solution of the scalar wave equation. Potential applicability of such simulation to data analysis and interpretation is demonstrated by comparing synthetic 3-D (x, y, t) common‐source gathers with those recorded on an areal grid of receivers over a physical scale model of a salt tongue. The iterative model adjustment process in fitting the physical model data is analogous to that which would be required for field data. Comparing differences between 2-D and 3-D synthetic responses allows interpretation of out‐of‐plane propagation. It is necessary to include the effects on the source directivity, because of a shield placed over the source during data acquisition, to numerically approximate the main features in the recorded scale model data.

An optimal frequency-domain finite-difference operator with a flexible stencil and its application in discontinuous-grid modeling

Geophysics ◽  
2021 ◽  
pp. 1-41
Author(s):  
Na Fan ◽  
Xiao-Bi Xie ◽  
Lian-Feng Zhao ◽  
Xin-Gong Tang ◽  
Zhen-Xing Yao
Keyword(s):  
Finite Difference ◽  
Frequency Domain ◽  
Waveform Inversion ◽  
Uniform Grid ◽  
Optimal Method ◽  
Velocity Models ◽  
Full Waveform ◽  
Rotationally Symmetric ◽  
Grid Points ◽  
Coarse To Fine

We develop an optimal method to determine expansion parameters for flexible stencils in 2D scalar wave finite-difference frequency-domain (FDFD) simulation. The proposed stencil only requires the involved grid points to be paired and rotationally symmetric around the central point. We apply this method to the transition zone in discontinuous-grid modeling, where the key issue is designing particular FDFD stencils to correctly propagate the wavefield passing through the discontinuous interface. The proposed method can work in FDFD discontinuous-grid with arbitrary integer coarse-to-fine gird spacing ratios. Numerical examples are presented to demonstrate how to apply this optimal method for the discontinuous-grid FDFD schemes with spacing ratios 3 and 5. The synthetic wavefields are highly consistent to those calculated using the conventional dense uniform grid, while the memory requirement and computational costs are greatly reduced. For velocity models with large contrasts, the proposed discontinuous-grid FDFD method can significantly improve the computational efficiency in forward modeling, imaging and full waveform inversion.

Forward Modeling of GPR Data by Unstaggered Finite Difference Frequency Domain (FDFD) Method: An Approach towards an Appropriate Numerical Scheme

2019 ◽  
Vol 24 (3) ◽  
pp. 487-496
Author(s):  
Mrinal Kanti Layek ◽  
Probal Sengupta
Keyword(s):  
Finite Difference ◽  
Frequency Domain ◽  
Homogeneous Model ◽  
Fdtd Method ◽  
Forward Modeling ◽  
Difference Frequency ◽  
Complex Frequency ◽  
Numerical Schemes ◽  
Finite Difference Frequency Domain ◽  
Ground Penetrating

Forward modeling of ground penetrating radar (GPR) is an important part to the inversion/modeling of the observed data. The aim of this study is to establish specific numerical schemes for forward modeling of GPR data by finite difference frequency domain (FDFD) method which were originally developed for seismic or finite difference time domain (FDTD) method. A total number of six modified and improved FDFD techniques have been used to discretize the two-dimensional (2D) transverse electric (TE)-mode scalar wave equation in order to find the suitable method for this. These techniques include five-point classical to nine-point mixed unstaggered-grid configurations. The numerical schemes for three unsplit perfectly matched layer (PML) for nine-point mixed unstaggered-grid configurations are also presented. The applicability of these techniques is tested by using the underground models of relative permittivity and conductivity for the two cases of homogeneous and 2-cross models. GPR shot gather data for these two models are also produced for this study. The relative reflection errors of the numerical schemes are also estimated for the homogeneous model to comprehend the appropriate method for the modeling. The algorithm for complex-frequency shifted PML (CFSPML) gives the least error in case of the forward modeling of the GPR data.

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