Iterative trace deconvolution and noncausal transform for processing band‐limited data

Geophysics ◽  
1990 ◽  
Vol 55 (12) ◽  
pp. 1549-1557 ◽  
Author(s):  
Philip Carrion ◽  
A. de Pinto Braga
Keyword(s):  
Acoustic Impedance ◽  
Real Data ◽  
Limited Data ◽  
Low Frequencies ◽  
True Solution ◽  
Integration Schemes ◽  
Ill Posed ◽  
Band Limited ◽  
Small Eigenvalues ◽  
Spectrum Inversion

In 1983, Carrion and Patton showed that if recorded seismic data do not have low frequencies in their spectrum, inversion for acoustic impedance is unstable. This concept is commonly accepted. Here, we infer acoustic impedance from band‐limited data without low frequencies using iterative trace deconvolution with the so‐called “noncausal projection” which moves initial data samples corresponding to small eigenvalues to negative times. We show that this procedure can solve ill‐posed problems by migrating large residuals (uncertainties in the trend of the recovered impedance) to the bottom of the model. An important property of the proposed method is that it converges to the true solution independently of the chosen initial model. Another advantage of the proposed algorithm is that, unlike conventional dynamic deconvolution (characteristic‐integration schemes), it increases neither errors nor the condition number with depth. The method is illustrated on synthetic and real data.

Feasibility of approximate broadband estimation of acoustic impedance profile from first principles and band-limited reflection data

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. R57-R74 ◽  
Author(s):  
Santi Kumar Ghosh ◽  
Animesh Mandal
Keyword(s):  
First Principles ◽  
Acoustic Impedance ◽  
Linear Equations ◽  
Synthetic Data ◽  
Reflection Coefficients ◽  
Limited Data ◽  
Seismic Reflection Data ◽  
Reflection Data ◽  
Impedance Profile ◽  
Band Limited

Because seismic reflection data are band limited, acoustic impedance profiles derived from them are nonunique. The conventional inversion methods counter the nonuniqueness either by stabilizing the answer with respect to an initial model or by imposing mathematical constraints such as sparsity of the reflection coefficients. By making a nominal assumption of an earth model locally consisting of a stack of homogeneous and horizontal layers, we have formulated a set of linear equations in which the reflection coefficients are the unknowns and the recursively integrated seismic trace constitute the data. Drawing only on first principles, the Zoeppritz equation in this case, the approach makes a frontal assault on the problem of reconstructing reflection coefficients from band-limited data. The local layer-cake assumption and the strategy of seeking a singular value decomposition solution of the linear equations counter the nonuniqueness, provided that the objective is to reconstruct a smooth version of the impedance profile that includes only its crude structures. Tests on synthetic data generated from elementary models and from measured logs of acoustic impedance demonstrated the efficacy of the method, even when a significant amount of noise was added to the data. The emergence of consistent estimates of impedance, approximating the original impedance, from synthetic data generated for several frequency bands has inspired our confidence in the method. The other attractive outputs of the method are as follows: (1) an accurate estimate of the impedance mean, (2) an accurate reconstruction of the direct-current (DC) frequency of the reflectivity, and (3) an acceptable reconstruction of the broad outline of the original impedance profile. These outputs can serve as constraints for either more refined inversions or geologic interpretations. Beginning from the restriction of band-limited data, we have devised a method that neither requires a starting input model nor imposes mathematical constraints on the earth reflectivity and still yielded significant and relevant geologic information.

Constrained full-waveform inversion by model reparameterization

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. R117-R127 ◽  
Author(s):  
Antoine Guitton ◽  
Gboyega Ayeni ◽  
Esteban Díaz
Keyword(s):  
Waveform Inversion ◽  
Model Space ◽  
Real Data ◽  
Velocity Model ◽  
Full Waveform Inversion ◽  
Coherent Noise ◽  
Low Frequencies ◽  
Full Waveform ◽  
Geological Information ◽  
Ill Posed

The waveform inversion problem is inherently ill-posed. Traditionally, regularization schemes are used to address this issue. For waveform inversion, where the model is expected to have many details reflecting the physical properties of the Earth, regularization and data fitting can work in opposite directions: the former smoothing and the latter adding details to the model. We propose constraining estimated velocity fields by reparameterizing the model. This technique, also called model-space preconditioning, is based on directional Laplacian filters: It preserves most of the details of the velocity model while smoothing the solution along known geological dips. Preconditioning also yields faster convergence at early iterations. The Laplacian filters have the property to smooth or kill local planar events according to a local dip field. By construction, these filters can be inverted and used in a preconditioned waveform inversion strategy to yield geologically meaningful models. We illustrate with 2D synthetic and field data examples how preconditioning with nonstationary directional Laplacian filters outperforms traditional waveform inversion when sparse data are inverted and when sharp velocity contrasts are present. Adding geological information with preconditioning could benefit full-waveform inversion of real data whenever irregular geometry, coherent noise and lack of low frequencies are present.

On a modified algorithm for the autoregressive recovery of the acoustic impedance

Geophysics ◽  
1984 ◽  
Vol 49 (12) ◽  
pp. 2190-2192 ◽  
Author(s):  
Tad. J. Ulrych ◽  
Colin Walker
Keyword(s):  
Acoustic Impedance ◽  
Low Frequency ◽  
Prediction Errors ◽  
Low Frequencies ◽  
Limited Frequency ◽  
Band Limited ◽  
Ar Modeling ◽  
Ar Process ◽  
Backward Prediction ◽  
Zero Frequency

In a recent paper, Walker and Ulrych (1983) presented an algorithm for the recovery of the acoustic impedance from band‐limited seismic reflection records. The approach used is based on the autoregressive (AR) modeling of the band‐limited frequency transform of the data. This modeling procedure allows prediction of both the high and low missing frequencies. The low frequencies, which are particularly important in the inversion for the acoustic impedance, are determined by considering the low‐frequency band as a gap of missing data which is centered at zero frequency. The gap is filled by minimizing the sum of the squared forward and backward prediction errors which result when the known spectral data are modeled as an AR process.

Recovery of the acoustic impedance from reflection seismograms

Geophysics ◽  
1983 ◽  
Vol 48 (10) ◽  
pp. 1318-1337 ◽  
Author(s):  
D. W. Oldenburg ◽  
T. Scheuer ◽  
S. Levy
Keyword(s):  
Acoustic Impedance ◽  
Recursion Formula ◽  
Low Frequency ◽  
Normal Incidence ◽  
Computationally Efficient ◽  
Low Frequencies ◽  
Frequency Information ◽  
Reflectivity Function ◽  
Band Limited ◽  
Common Signal

This paper examines the problem of recovering the acoustic impedance from a band‐limited normal incidence reflection seismogram. The convolutional model for the seismogram is adopted at the outset, and it is therefore required that initial processing has removed multiples and recovered true amplitudes as well as possible. In the first portion of the paper we investigate the effect of substituting the deconvolved seismic trace (that is, the band‐limited version of the reflectivity function) into the standard recursion formula for the acoustic impedance. The formalism of linear inverse theory is used to show that the logarithm of the normalized acoustic impedance estimated from the deconvolved seismogram is approximately an average of the true logarithm of the impedance. Moreover, the averaging function is identical to that used in deconvolving the initial seismogram. The advantage of these averages is that they are unique; their disadvantage is that low‐frequency information, which is crucial to making a geologic interpretation, is missing. We next present two methods by which the missing low‐frequency information can be recovered. The first method is a linear programming (LP) construction algorithm which attempts to find a reflectivity function made of isolated delta functions. This method is computationally efficient and robust in the presence of noise. Importantly, it also lends itself to the incorporation of impedance constraints if such geologic information is available. A second construction method makes use of the fact that the Fourier transform of a reflectivity function for a layered earth can be modeled as an autoregressive (AR) process. The missing high and low frequencies can thus be predicted from the band‐limited reflectivity function by standard techniques. Stability in the presence of additive noise on the seismogram is achieved by predicting frequencies outside the known frequency band with operators of different orders and extracting a common signal from the results. Our construction algorithms are shown to operate successfully on a variety of synthetic examples. Two sections of field data are inverted, and in both the results from the LP and AR methods are similar and compare favorably to acoustic impedance features observed at nearby wells.

Acoustic Neuromas and the Acoustic Impedance of the Ear

1973 ◽  
Vol 38 (3) ◽  
pp. 345-353 ◽  
Author(s):  
J. H. Macrae
Keyword(s):  
Tympanic Membrane ◽  
Acoustic Neuroma ◽  
Acoustic Impedance ◽  
Theoretical Calculations ◽  
Low Frequencies ◽  
Cochlear Duct ◽  
Acoustic Neuromas ◽  
Affected Side ◽  
Contralateral Ear ◽  
Conductive System

The acoustic impedance at the tympanic membrane was measured at frequencies in the range 100–1000 Hz and found to be abnormal on the affected side in four patients with acoustic neuroma. In all four the resistance was abnormally high at low frequencies on the affected side, and in three the reactance of the affected ear was raised relative to that of the contralateral ear, particularly at low frequencies. The abnormality is attributed to an increase in the input acoustic impedance of the cochlea produced by the increase in protein content of the cochlear fluids and dilatation of the cochlear duct known to occur in acoustic neuroma. This explanation is supported by theoretical calculations carried out on an electric analogue of the conductive system, and it is suggested that similar abnormalities in the acoustic impedance at the tympanic membrane might occur in other pathologies which produce abnormal mechanical conditions in the cochlea.

Robust wavelet estimation and blind deconvolution of noisy surface seismics

Geophysics ◽  
2008 ◽  
Vol 73 (5) ◽  
pp. V37-V46 ◽  
Author(s):  
Mirko van der Baan ◽  
Dinh-Tuan Pham
Keyword(s):  
Blind Deconvolution ◽  
General Purpose ◽  
Sufficient Accuracy ◽  
Wiener Filtering ◽  
Challenging Problem ◽  
Limited Data ◽  
Wavelet Estimation ◽  
Seismic Wavelet ◽  
Principal Frequency ◽  
Band Limited

Robust blind deconvolution is a challenging problem, particularly if the bandwidth of the seismic wavelet is narrow to very narrow; that is, if the wavelet bandwidth is similar to its principal frequency. The main problem is to estimate the phase of the wavelet with sufficient accuracy. The mutual information rate is a general-purpose criterion to measure whiteness using statistics of all orders. We modified this criterion to measure robustly the amplitude and phase spectrum of the wavelet in the presence of noise. No minimum phase assumptions were made. After wavelet estimation, we obtained an optimal deconvolution output using Wiener filtering. The new procedure performs well, even for very band-limited data; and it produces frequency-dependent phase estimates.

Colored and linear inversions to relative acoustic impedance

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. N15-N27 ◽  
Author(s):  
Carlos A. M. Assis ◽  
Henrique B. Santos ◽  
Jörg Schleicher
Keyword(s):  
Seismic Data ◽  
Acoustic Impedance ◽  
Computational Cost ◽  
Synthetic Data ◽  
Seismic Source ◽  
Alternative Methods ◽  
Well Log ◽  
Linear Inversion ◽  
Amplitude Inversion ◽  
Band Limited

Acoustic impedance (AI) is a widely used seismic attribute in stratigraphic interpretation. Because of the frequency-band-limited nature of seismic data, seismic amplitude inversion cannot determine AI itself, but it can only provide an estimate of its variations, the relative AI (RAI). We have revisited and compared two alternative methods to transform stacked seismic data into RAI. One is colored inversion (CI), which requires well-log information, and the other is linear inversion (LI), which requires knowledge of the seismic source wavelet. We start by formulating the two approaches in a theoretically comparable manner. This allows us to conclude that both procedures are theoretically equivalent. We proceed to check whether the use of the CI results as the initial solution for LI can improve the RAI estimation. In our experiments, combining CI and LI cannot provide superior RAI results to those produced by each approach applied individually. Then, we analyze the LI performance with two distinct solvers for the associated linear system. Moreover, we investigate the sensitivity of both methods regarding the frequency content present in synthetic data. The numerical tests using the Marmousi2 model demonstrate that the CI and LI techniques can provide an RAI estimate of similar accuracy. A field-data example confirms the analysis using synthetic-data experiments. Our investigations confirm the theoretical and practical similarities of CI and LI regardless of the numerical strategy used in LI. An important result of our tests is that an increase in the low-frequency gap in the data leads to slightly deteriorated CI quality. In this case, LI required more iterations for the conjugate-gradient least-squares solver, but the final results were not much affected. Both methodologies provided interesting RAI profiles compared with well-log data, at low computational cost and with a simple parameterization.

Multidimensional de-aliased Cadzow reconstruction of seismic records

Geophysics ◽  
2013 ◽  
Vol 78 (1) ◽  
pp. A1-A5 ◽  
Author(s):  
Mostafa Naghizadeh ◽  
Mauricio Sacchi
Keyword(s):  
Seismic Data ◽  
Interpolation Method ◽  
Hankel Matrix ◽  
Real Data ◽  
Low Frequencies ◽  
Frequency Space ◽  
Reduced Rank ◽  
Seismic Records ◽  
Text Prediction ◽  
Eigen Decomposition

We tested a strategy for beyond-alias interpolation of seismic data using Cadzow reconstruction. The strategy enables Cadzow reconstruction to be used for interpolation of regularly sampled seismic records. First, in the frequency-space ([Formula: see text]) domain, we generated a Hankel matrix from the spatial samples of the low frequencies. To perform interpolation at a given frequency, the spatial samples were interlaced with zero samples and another Hankel matrix was generated from the zero-interlaced data. Next, the rank-reduced eigen-decomposition of the Hankel matrix at low frequencies was used for beyond-alias preconditioning of the Hankel matrix at a given frequency. Finally, antidiagonal averaging of the conditioned Hankel matrix produced the final interpolated data. In addition, the multidimensional extension of the proposed algorithm was explained. The proposed method provides a unifying thread between reduced-rank Cadzow reconstruction and beyond alias [Formula: see text] prediction error interpolation. Synthetic and real data examples were provided to examine the performance of the proposed interpolation method.

A fast and simple method of spectral enhancement

Geophysics ◽  
2014 ◽  
Vol 79 (3) ◽  
pp. V75-V80 ◽  
Author(s):  
Muhammad Sajid ◽  
Deva Ghosh
Keyword(s):  
Seismic Data ◽  
Real Data ◽  
Difference Operators ◽  
Frequency Noise ◽  
Simple Method ◽  
Low Frequencies ◽  
High Frequencies ◽  
Algorithm Comparison ◽  
Spectral Enhancement ◽  
Bed Thickness

The ability to resolve seismic thin beds is a function of the bed thickness and the frequency content of the seismic data. To achieve high resolution, the seismic data must have broad frequency bandwidth. We developed an algorithm that improved the bandwidth of the seismic data without greatly boosting high-frequency noise. The algorithm employed a set of three cascaded difference operators to boost high frequencies and combined with a simple smoothing operator to boost low frequencies. The output of these operators was balanced and added to the original signal to produce whitened data. The four convolutional operators were quite short, so the algorithm was highly efficient. Synthetic and real data examples demonstrated the effectiveness of this algorithm. Comparison with a conventional whitening algorithm showed the algorithm to be competitive.

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