Nonlinear impedance inversion for attenuating media

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. R111-R117 ◽  
Author(s):  
Sérgio Oliveira ◽  
Luiz Loures ◽  
Fernando Moraes ◽  
Carlos Theodoro
Keyword(s):  
Amplitude Spectrum ◽  
Real Data ◽  
Alternative Formulation ◽  
Nonlinear Inversion ◽  
Inversion Algorithm ◽  
Impedance Inversion ◽  
Reflectivity Function ◽  
Analytical Formulas ◽  
Zero Offset ◽  
Internal Multiples

Applications of seismic impedance inversion normally assume the data are free of multiples and transmission effects, requiring knowledge of the seismic pulse that is assumed to be stationary. An alternative formulation for impedance inversion is based on an exact frequency-domain, zero-offset reflectivity function for a 1D medium. Analytical formulas for the Fréchet derivatives are derived for efficient implementation of an iterative nonlinear inversion. The exact zero-offset reflectivity accounts for internal multiples and transmission effects in the data. Absorption and dispersion are also conveniently handled if a reasonable estimate for the quality [Formula: see text] factor of the medium is available. A series of convenient features are included in the inversion algorithm: an automatic estimation of the amplitude spectrum of the seismic pulse, an impedance transform that makes the inversion independent from the initial smooth model, and a practical approach to estimate the regularization weight. Numerical tests using synthetic and real data show that the method is stable and needs only a few iterations to converge.

3D marine magnetotelluric inversion: A hybrid impedance and direct-field approach

Geophysics ◽  
2017 ◽  
Vol 82 (6) ◽  
pp. E335-E346
Author(s):  
Lutz Mütschard ◽  
Ketil Hokstad ◽  
Torgeir Wiik ◽  
Bjørn Ursin
Keyword(s):  
Electric Field ◽  
Real Data ◽  
Source Distribution ◽  
Impedance Tensor ◽  
Inversion Algorithm ◽  
Data Set ◽  
Wave Source ◽  
Field Approach ◽  
Impedance Inversion ◽  
Field Inversion

The measured electromagnetic field in magnetotellurics (MT) is composed of the natural source field and its subsurface response. Commonly, the data are represented as impedances, the complex ratio between the horizontal electric and magnetic fields. This measure is independent of the source distribution because the impedance-tensor estimation contains a deconvolution operator. We have used a Gauss-Newton-type 3D MT inversion scheme to compare impedance-data inversion with an inversion using the recorded electric field directly. The use of the observed electric field is beneficial to the inversion algorithm because it simplifies the estimation of the sensitivities. The direct-field approach permits the use of the observed data without processing, but it presumes knowledge of the source distribution. A method to estimate the time-variable strength and polarization of the incoming plane-wave source is presented and tested on synthetic and real-data examples. The direct-field inversion is successfully applied to a synthetic and a real data set within marine settings. A comparison with the conventional impedance inversion is conducted. The results of the synthetic data example are very similar, with a slightly more accurate reconstruction of the model in the impedance case, whereas the direct-field inversion produces a smoother inversion result when compared with the impedance case. The mapping of a resistive salt structure in the real-data example indicates deviations in the final conductivity models. The impedance inversion suggests a deeper rooted resistive structure, whereas the direct-field inversion predicts a more compact structure limited to the overburden. We have evaluated the advantages of the new approach like the simplification of the sensitivity calculation, limitations, and disadvantages like knowledge of the source distribution.

Large‐scale three‐dimensional seismic models and their interpretive significance

Geophysics ◽  
1990 ◽  
Vol 55 (9) ◽  
pp. 1166-1182 ◽  
Author(s):  
Irshad R. Mufti
Keyword(s):  
Finite Difference ◽  
Wave Field ◽  
Large Scale ◽  
Three Dimensional ◽  
Synthetic Data ◽  
Real Data ◽  
The Real ◽  
Need To Evaluate ◽  
Zero Offset ◽  
Seismic Models

Finite‐difference seismic models are commonly set up in 2-D space. Such models must be excited by a line source which leads to different amplitudes than those in the real data commonly generated from a point source. Moreover, there is no provision for any out‐of‐plane events. These problems can be eliminated by using 3-D finite‐difference models. The fundamental strategy in designing efficient 3-D models is to minimize computational work without sacrificing accuracy. This was accomplished by using a (4,2) differencing operator which ensures the accuracy of much larger operators but requires many fewer numerical operations as well as significantly reduced manipulation of data in the computer memory. Such a choice also simplifies the problem of evaluating the wave field near the subsurface boundaries of the model where large operators cannot be used. We also exploited the fact that, unlike the real data, the synthetic data are free from ambient noise; consequently, one can retain sufficient resolution in the results by optimizing the frequency content of the source signal. Further computational efficiency was achieved by using the concept of the exploding reflector which yields zero‐offset seismic sections without the need to evaluate the wave field for individual shot locations. These considerations opened up the possibility of carrying out a complete synthetic 3-D survey on a supercomputer to investigate the seismic response of a large‐scale structure located in Oklahoma. The analysis of results done on a geophysical workstation provides new insight regarding the role of interference and diffraction in the interpretation of seismic data.

2-D continuation operators and their applications

Geophysics ◽  
1996 ◽  
Vol 61 (6) ◽  
pp. 1846-1858 ◽  
Author(s):  
Claudio Bagaini ◽  
Umberto Spagnolini
Keyword(s):  
Seismic Data ◽  
Real Data ◽  
Integral Form ◽  
Amplitude Distribution ◽  
Velocity Model ◽  
Velocity Analysis ◽  
Seismic Data Processing ◽  
Analysis Method ◽  
Standard Tool ◽  
Zero Offset

Continuation to zero offset [better known as dip moveout (DMO)] is a standard tool for seismic data processing. In this paper, the concept of DMO is extended by introducing a set of operators: the continuation operators. These operators, which are implemented in integral form with a defined amplitude distribution, perform the mapping between common shot or common offset gathers for a given velocity model. The application of the shot continuation operator for dip‐independent velocity analysis allows a direct implementation in the acquisition domain by exploiting the comparison between real data and data continued in the shot domain. Shot and offset continuation allow the restoration of missing shot or missing offset by using a velocity model provided by common shot velocity analysis or another dip‐independent velocity analysis method.

Frequency‐domain acoustic‐wave modeling and inversion of crosshole data: Part II—Inversion method, synthetic experiments and real‐data results

Geophysics ◽  
1995 ◽  
Vol 60 (3) ◽  
pp. 796-809 ◽  
Author(s):  
Zhong‐Min Song ◽  
Paul R. Williamson ◽  
R. Gerhard Pratt
Keyword(s):  
Frequency Domain ◽  
Real Data ◽  
Forward Modeling ◽  
Two Dimensions ◽  
Inversion Method ◽  
Inversion Algorithm ◽  
Data Set ◽  
Wave Inversion ◽  
Full Wave ◽  
And Inversion

In full‐wave inversion of seismic data in complex media it is desirable to use finite differences or finite elements for the forward modeling, but such methods are still prohibitively expensive when implemented in 3-D. Full‐wave 2-D inversion schemes are of limited utility even in 2-D media because they do not model 3-D dynamics correctly. Many seismic experiments effectively assume that the geology varies in two dimensions only but generate 3-D (point source) wavefields; that is, they are “two‐and‐one‐half‐dimensional” (2.5-D), and this configuration can be exploited to model 3-D propagation efficiently in such media. We propose a frequency domain full‐wave inversion algorithm which uses a 2.5-D finite difference forward modeling method. The calculated seismogram can be compared directly with real data, which allows the inversion to be iterated. We use a descents‐related method to minimize a least‐squares measure of the wavefield mismatch at the receivers. The acute nonlinearity caused by phase‐wrapping, which corresponds to time‐domain cycle‐skipping, is avoided by the strategy of either starting the inversion using a low frequency component of the data or constructing a starting model using traveltime tomography. The inversion proceeds by stages at successively higher frequencies across the observed bandwidth. The frequency domain is particularly efficient for crosshole configurations and also allows easy incorporation of attenuation, via complex velocities, in both forward modeling and inversion. This also requires the introduction of complex source amplitudes into the inversion as additional unknowns. Synthetic studies show that the iterative scheme enables us to achieve the theoretical maximum resolution for the velocity reconstruction and that strongly attenuative zones can be recovered with reasonable accuracy. Preliminary results from the application of the method to a real data set are also encouraging.

Processing via spectral modeling

Geophysics ◽  
1991 ◽  
Vol 56 (8) ◽  
pp. 1244-1251 ◽  
Author(s):  
A. L. R. Rosa ◽  
T. J. Ulrych
Keyword(s):  
Real Data ◽  
Important Conclusion ◽  
Spectral Modeling ◽  
Processing Scheme ◽  
Widespread Occurrence ◽  
Second Stage ◽  
Reflectivity Function ◽  
Color Compensation ◽  
Two Stages ◽  
Subtle Trap

The widespread occurrence of subtle trap accumulations offshore Brazil has led to the need for the development of a high resolution processing scheme that helps the delineation of these features. The process consists of three stages, the first of which is deterministic and stochastic deconvolution. The second stage is the deconvolution of the residual wavelet by means of spectral modeling. The last stage consists of the correction of the color of the reflectivity function using a model developed for the area. An important conclusion that is drawn from the model is that the acoustic impedance is not white. Rather it is as red as the corresponding reflectivity is blue. Successful results from the application of the proposed technique to real data indicate that the color compensation is of second order importance as compared with the first two stages of the proposed scheme.

Turning‐ray tomography

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1917-1929 ◽  
Author(s):  
Joseph P. Stefani
Keyword(s):  
Velocity Structure ◽  
Initial Point ◽  
Real Data ◽  
Point Sources ◽  
Surface Velocity ◽  
Inversion Algorithm ◽  
Low Velocity ◽  
Linear Inversion ◽  
Near Surface ◽  
Velocity Inversion

Turning‐ray tomography is useful for estimating near‐surface velocity structure in areas where conventional refraction statics techniques fail because of poor data or lack of smooth refractor/velocity structure. This paper explores the accuracy and inherent smoothing of turning‐ray tomography in its capacity to estimate absolute near‐surface velocity and the statics times derived from these velocities, and the fidelity with which wavefields collapse to point diffractors when migrated through these estimated velocities. The method comprises nonlinear iterations of forward ray tracing through triangular cells linear in slowness squared, coupled with the LSQR linear inversion algorithm. It is applied to two synthetic finite‐ difference data sets of types that usually foil conventional refraction statics techniques. These models represent a complex hard‐rock overthrust structure with a low‐velocity zone and pinchouts, and a contemporaneous near‐shore marine trench filled with low‐ velocity unconsolidated deposits exhibiting no seismically apparent internal structure. In both cases velocities are estimated accurately to a depth of one‐ fifth the maximum offset, as are the associated statics times. Of equal importance, the velocities are sufficiently accurate to correctly focus synthetic wavefields back to their initial point sources, so migration/datuming applications can also use these velocities. The method is applied to a real data example from the Timbalier Trench in the Gulf of Mexico, which exhibits the same essential features as the marine trench synthetic model. The Timbalier velocity inversion is geologically reasonable and yields long and short wavelength statics that improve the CMP gathers and stack and that correctly align reflections to known well markers. Turning‐ray tomography estimates near‐surface velocities accurately enough for the three purposes of lithology interpretation, statics calculations, and wavefield focusing for shallow migration and datuming.

A generic 1-D imaging method for transient electromagnetic data

Geophysics ◽  
2002 ◽  
Vol 67 (2) ◽  
pp. 438-447 ◽  
Author(s):  
Niels Bøie Christensen
Keyword(s):  
Time Derivative ◽  
Step Response ◽  
Imaging Method ◽  
Nonlinear Inversion ◽  
Inversion Algorithm ◽  
Electromagnetic Data ◽  
Transient Electromagnetic ◽  
Dependent Part ◽  
Space Step ◽  
Forward Mapping

This paper presents a fast approximate 1-D inversion algorithm for transient electromagnetic (EM) data that can be applied for all measuring configurationsand transmitter waveforms and for all field components. The inversion is based on an approximate forward mapping in the adaptive Born approximation. The generality is obtained through a separation of the forward problem into a configuration-independent part, mapping layer conductivities into apparent conductivity, and a configuration-dependent part, the half-space step response. The EM response from any waveform can then be found by a convolution with the time derivative of the waveform. The approach does not involve inherently unstable deconvolution computations or nonunique transformations, and it is about 100 times faster than ordinary nonlinear inversion. Nonlinear model responses of the models obtained through the approximate inversion fit the data typically within 5%.

AVA analysis after velocity-independent DMO and imaging

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 686-691 ◽  
Author(s):  
Gerald H. F. Gardner ◽  
Anat Canning
Keyword(s):  
Reflection Coefficient ◽  
Peak Amplitude ◽  
Angle Of Incidence ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
The Past ◽  
Reflectivity Function ◽  
Velocity Of Propagation ◽  
Zero Offset ◽  
Amplitude Versus

A common midpoint (CMP) gather usually provides amplitude variation with offset (AVO) information by displaying the reflectivity as the peak amplitude of symmetrical deconvolved wavelets. This puts a reflection coefficient R at every offset h, giving a function R(h). But how do we link h with the angle of incidence, θ, to get the reflectivity function, R(θ)? This is necessary for amplitude versus angle-of-incidence (AVA) analysis. One purpose of this paper is to derive formulas for this linkage after velocity-independent dip-moveout (DMO), done by migrating radial sections, and prestack zero-offset migration. Related studies of amplitude-preserving DMO in the past have dealt with constant-offset DMO but have not given the connection between offset and angle of incidence after processing. The results in the present paper show that the same reflectivity function can be extracted from the imaged volume whether it is produced using radial-trace DMO plus zero-offset migration, constant-offset DMO plus zero-offset migration, or directly by prestack, common-offset migration. The data acquisition geometry for this study consists of parallel, regularly spaced, multifold lines, and the velocity of propagation is constant. Events in the data are caused by an arbitrarily oriented 3-D plane reflector with any reflectivity function. The DMO operation transforms each line of data (m, h, t), i.e., midpoint, half-offset, and time, into an (m1, k, t1) space by Stolt-migrating each radial-plane section of the data, 2h = Ut, with constant velocity U/2. Merging the (m1, k, t1) spaces for all the lines forms an (x, y, k, t1) space, where the first two coordinates are the midpoint location, the third is the new half-offset, and the fourth is the time. Normal moveout (NMO) plus 3-D zero-offset migration of the subspace (x, y, t1) for each k creates a true-amplitude imaged volume (X, Y, k, T). Each peak amplitude in the volume is a reflection coefficient linked to an angle of incidence.

Thin-bed prestack spectral inversion

Geophysics ◽  
2009 ◽  
Vol 74 (4) ◽  
pp. R49-R57 ◽  
Author(s):  
J. Germán Rubino ◽  
Danilo Velis
Keyword(s):  
Time Window ◽  
A Priori ◽  
Amplitude Spectrum ◽  
Wavelet Spectrum ◽  
Model Parameters ◽  
Data Sets ◽  
S Wave ◽  
P Waves ◽  
S Waves ◽  
Zero Offset

Prestack seismic data has been used in a new method to fully determine thin-bed properties, including the estimation of its thickness, P- and S-wave velocities, and density. The approach requires neither phase information nor normal-moveout (NMO) corrections, and assumes that the prestack seismic response of the thin layer can be isolated using an offset-dependent time window. We obtained the amplitude-versus-angle (AVA) response of the thin bed considering converted P-waves, S-waves, and all the associated multiples. We carried out the estimation of the thin-bed parameters in the frequency (amplitude spectrum) domain using simulated annealing. In contrast to using zero-offset data, the use of AVA data contributes to increase the robustness of this inverse problem under noisy conditions, as well as to significantly reduce its inherent nonuniqueness. To further reduce the nonuniqueness, and as a means to incorporate a priori geologic or geophysical information (e.g., well-log data), we imposed appropriate bounding constraints to the parameters of the media lying above and below the thin bed, which need not be known accurately. We tested the method by inverting noisy synthetic gathers corresponding to simple wedge models. In addition, we stochastically estimated the uncertainty of the solutions by inverting different data sets that share the same model parameters but are contaminated with different noise realizations. The results suggest that thin beds can be characterized fully with a moderate to high degree of confidence below tuning, even when using an approximate wavelet spectrum.

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