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.

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.

Low Frequency Absorption of Additively Manufactured Cylinders

2019 ◽  
Author(s):  
Sophie R. Kaye ◽  
Ethan D. Casavant ◽  
Paul E. Slaboch
Keyword(s):  
Low Frequency ◽  
Normal Incidence ◽  
Frequency Noise ◽  
Outer Diameter ◽  
Acoustic Absorption ◽  
Frequency Range ◽  
Large Space ◽  
Low Frequencies ◽  
Cylinder Length ◽  
Hollow Cylinders

Abstract Attenuating low frequencies is often problematic, due to the large space required for common absorptive materials to mitigate such noise. However, natural hollow reeds are known to effectively attenuate low frequencies while occupying relatively little space compared to traditional absorptive materials. This paper discusses the effect of varied outer diameter, and outer spacing on the 200–1600 Hz acoustic absorption of additively manufactured arrays of hollow cylinders. Samples were tested in a 10 cm diameter normal incidence impedance tube such that cylinder length was oriented perpendicular to the incoming plane wave. By varying only one geometric element of each array, the absorption due to any particular parameter can be assessed individually. The tests confirmed the hypothesis that minimizing cylinder spacing and maximizing cylinder diameter resulted in increased overall absorption and produced more focused absorption peaks at specific low frequencies. Wider cylinder spacing produced a broader absorptive frequency range, despite shifting upward in frequency. Thus, manipulating these variables can specifically target absorption for low frequency noise that would otherwise disturb listeners.

Control-Oriented Modeling of Lithium-Ion Batteries

2020 ◽  
Author(s):  
Brody Riemann ◽  
Jie Li ◽  
Kasim Adewuyi ◽  
Robert Landers ◽  
Jonghyun Park
Keyword(s):  
Low Frequency ◽  
Lithium Ion ◽  
Battery Management ◽  
Computationally Efficient ◽  
Low Frequencies ◽  
Battery Modeling ◽  
Capacitance Effect ◽  
Frequency Domains ◽  
Frequency Behavior ◽  
Time And Frequency Domains

Abstract Battery Management Systems (BMSs) require control-oriented models. Physics-based electrochemical models describe detailed battery phenomena, but are too computationally intensive for use in BMSs. Single Particle Models (SPMs) are often used for control-oriented battery modeling since they are physics-based and computationally efficient; however, they are only valid over very low frequency ranges and C-rates. Empirical Equivalent Circuit Models (ECMs) are also used in BMSs since they are computationally efficient and describe battery behavior over wide frequency ranges; however, they provide no physical understanding of the battery and often employ fractional order terms. This work provides a control-oriented battery model that combines the benefits of SPM and ECM models, while overcoming their limitations. The proposed model incorporates some of the battery physics found in electrochemical models, can easily be used in both the time and frequency domains, and describes battery behavior over its entire frequency range. A linearized SPM models battery physics at very low frequencies. For low frequencies, integer-order linear systems are used to approximate diffusion physics, and high frequency behavior is modeled by the double layer capacitance effect. The model is validated in the time and frequency domains via a comparison to Pseudo 2-Dimensional (P2D) model simulations and experimental data.

Interpretation of full-azimuth broadband land data from Saudi Arabia and implications for improved inversion, reservoir characterization, and exploration

2013 ◽  
Vol 1 (2) ◽  
pp. T167-T176 ◽  
Author(s):  
Brian P. Wallick ◽  
Luis Giroldi
Keyword(s):  
Saudi Arabia ◽  
Seismic Data ◽  
Acoustic Impedance ◽  
Reservoir Characterization ◽  
Signal To Noise Ratio ◽  
Low Frequency ◽  
Gas Field ◽  
Frequency Model ◽  
Low Frequencies ◽  
Limited Bandwidth

Interpretation of conventional land seismic data over a Permian-age gas field in Eastern Saudi Arabia has proven difficult over time due to low signal-to-noise ratio and limited bandwidth in the seismic volume. In an effort to improve the signal and broaden the bandwidth, newly acquired seismic data over this field have employed point receiver technology, dense wavefield sampling, a full azimuth geometry, and a specially designed sweep with useful frequencies as low as three hertz. The resulting data display enhanced reflection continuity and improved resolution. With the extension of low frequencies and improved interpretability, acoustic impedance inversion results are more robust and allow greater flexibility in reservoir characterization and prediction. In addition, because inversion to acoustic impedance is no longer completely tied to a wells-only low-frequency model, there are positive implications for exploration.

3D printed multifunctional, load-bearing, low-frequency sound absorbers

2021 ◽  
Vol 263 (1) ◽  
pp. 5605-5610
Author(s):  
William Johnston ◽  
Pulitha Godakawela Kankanamalage ◽  
Bhisham Sharma
Keyword(s):  
Mechanical Performance ◽  
Low Frequency ◽  
Normal Incidence ◽  
Acoustic Properties ◽  
Low Frequencies ◽  
Load Bearing ◽  
Fibrous Network ◽  
Size Limitation ◽  
Porous Architecture ◽  
3D Printed

Cellular porous materials are an attractive choice for lightweight structural design. However, though their open porous architecture is ideally suited for multifunctional applications, their use is typically limited by the pore sizes achievable by traditional as well as advanced fabrication processes. Here, we present an alternative route towards overcoming this pore size limitation by leveraging our recent success in printing fibrous structures. This is achieved by superimposing a fibrous network on a load-bearing, open-celled porous architecture. The multifunctional structure is 3D printed using a novel technique that enables us to simultaneously print a load-bearing scaffold and the necessary fibrous network. The acoustic properties of the printed structures are tested using a normal-incidence impedance tube method. Our results show that such structures can provide very high absorption at low frequencies while retaining the mechanical performance of the underlying architected structure.

Low-Frequency Acoustic Absorption of 3D Printed Cylinders

2020 ◽  
Vol 143 (1) ◽  
Author(s):  
Paul E. Slaboch ◽  
Sophie Kaye ◽  
Ethan Casavant
Keyword(s):  
Low Frequency ◽  
Normal Incidence ◽  
Frequency Noise ◽  
Outer Diameter ◽  
Acoustic Absorption ◽  
Large Space ◽  
Low Frequencies ◽  
Cylinder Length ◽  
3D Printed ◽  
Hollow Cylinders

Abstract Attenuating low-frequency sound is often problematic, due to the large space required for common absorptive materials to mitigate such noise. However, natural hollow reeds are known to effectively attenuate low frequencies while occupying relatively little space compared to traditional absorptive materials. The present study determines the effect of varied outer diameter and outer spacing on the 200–1600 Hz acoustic absorption of 3D printed arrays of hollow cylinders. Samples were tested in a 100-mm diameter normal incidence impedance tube such that cylinder length was oriented perpendicular to the incoming plane wave. By varying only one geometric element of each array, the absorption due to any parameter can be assessed individually. It was found that minimizing cylinder spacing and maximizing cylinder diameter resulted in increased overall absorption and produced more focused absorption peaks at specific low frequencies. Wider cylinder spacing produced a broader absorptive frequency range, despite shifting upward in frequency. Thus, manipulating these variables can specifically target absorption for low-frequency noise that would otherwise disturb listeners.

Acoustic impedance estimation from combined harmonic reconstruction and interval velocity

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. R385-R400
Author(s):  
Luca Bianchin ◽  
Emanuele Forte ◽  
Michele Pipan
Keyword(s):  
Velocity Field ◽  
Seismic Data ◽  
Acoustic Impedance ◽  
Low Frequency ◽  
Reconstruction Method ◽  
Well Log ◽  
Data Set ◽  
Low Frequencies ◽  
Interval Velocity ◽  
Frequency Components

Low-frequency components of reflection seismic data are of paramount importance for acoustic impedance (AI) inversion, but they typically suffer from a poor signal-to-noise ratio. The estimation of the low frequencies of AI can benefit from the combination of a harmonic reconstruction method (based on autoregressive [AR] models) and a seismic-derived interval velocity field. We have developed the construction of a convex cost function that accounts for the velocity field, together with geologic a priori information on AI and its uncertainty, during the AR reconstruction of the low frequencies. The minimization of this function allows one to reconstruct sensible estimates of low-frequency components of the subsurface reflectivity, which lead to an estimation of AI model via a recursive formulation. In particular, the method is suited for an initial and computationally inexpensive assessment of the absolute value of AI even when no well-log data are available. We first tested the method on layered synthetic models, then we analyzed its applicability and limitations on a real marine seismic data set that included tomographic velocity information. Despite a strong trace-to-trace variability in the results, which could partially be mitigated by multitrace inversion, the method demonstrates its capability to highlight lateral variations of AI that cannot be detected when the low frequencies only come from well-log information.

scholarly journals Extrapolated full-waveform inversion with deep learning

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. R275-R288 ◽  
Author(s):  
Hongyu Sun ◽  
Laurent Demanet
Keyword(s):  
Neural Network ◽  
Numerical Experiments ◽  
Waveform Inversion ◽  
Low Frequency ◽  
Full Waveform Inversion ◽  
Bandwidth Extension ◽  
Low Frequencies ◽  
Full Waveform ◽  
The Neural Network ◽  
Band Limited

The lack of low-frequency information and a good initial model can seriously affect the success of full-waveform inversion (FWI), due to the inherent cycle skipping problem. Computational low-frequency extrapolation is in principle the most direct way to address this issue. By considering bandwidth extension as a regression problem in machine learning, we have adopted an architecture of convolutional neural network (CNN) to automatically extrapolate the missing low frequencies. The band-limited recordings are the inputs of the CNN, and, in our numerical experiments, a neural network trained from enough samples can predict a reasonable approximation to the seismograms in the unobserved low-frequency band, in phase and in amplitude. The numerical experiments considered are set up on simulated P-wave data. In extrapolated FWI (EFWI), the low-wavenumber components of the model are determined from the extrapolated low frequencies, before proceeding with a frequency sweep of the band-limited data. The introduced deep-learning method of low-frequency extrapolation shows adequate generalizability for the initialization step of EFWI. Numerical examples show that the neural network trained on several submodels of the Marmousi model is able to predict the low frequencies for the BP 2004 benchmark model. Additionally, the neural network can robustly process seismic data with uncertainties due to the existence of random noise, a poorly known source wavelet, and a different finite-difference scheme in the forward modeling operator. Finally, this approach is not subject to strong assumptions on signals or velocity models of other methods for bandwidth extension and seems to offer a tantalizing solution to the problem of properly initializing FWI.

Prestack inversion with plane‐layer point source modeling

Geophysics ◽  
1985 ◽  
Vol 50 (1) ◽  
pp. 77-89 ◽  
Author(s):  
Alastair D. McAulay
Keyword(s):  
Velocity Profile ◽  
Point Source ◽  
Spherical Wave ◽  
Normal Incidence ◽  
Plane Layer ◽  
Source Modeling ◽  
Low Frequencies ◽  
Prestack Inversion ◽  
Band Limited

Prestack inversion with point‐source plane‐layer modeling has many advantages over poststack or normal incidence inversion. For example, it permits the determination of absolute compressional and shear velocities, density variations, and the accurate accounting of interbed and surface multiples. I neglect shear effects in this paper by assuming that they are adequately suppressed by velocity filtering. In the forward modeling step, a spherical wave expansion into plane waves is used to account for the point source. The plane‐wave reflection response for a set of plane layers is extended to the nonnormal incidence case. I use a Hankel transform to account for cylindrical symmetry. Generalized linear inversion is used because the fast recursive approaches available for normal incidence inversion are no longer applicable. I provide the derivation for the required derivative matrix, and I take into account the band‐limited nature of the data in frequency, time, and space. I demonstrate that moveout of events on realistic simulated prestack data enables the determination of absolute compressional velocity in the velocity‐depth profile, even though the data are band‐limited in frequency. I assume that preprocessing has adequately removed the shear and surface effects and that density is constant. Low frequencies in the velocity profile may be obtained more accurately than with velocity analysis used for stacking, because interbed multiples and other modeling phenomena are accounted for in the computation. Autoregressive modeling procedures that predict into the low frequencies of the velocity profile are also less accurate and cannot generate absolute velocity. I suggest future research leading to cost‐effective inversion of real data.

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.

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