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.

Autoregressive recovery of the acoustic impedance

Geophysics ◽  
1983 ◽  
Vol 48 (10) ◽  
pp. 1338-1350 ◽  
Author(s):  
Colin Walker ◽  
Tad J. Ulrych
Keyword(s):  
Frequency Band ◽  
Acoustic Impedance ◽  
High Frequency ◽  
Low Frequency ◽  
Minimum Entropy ◽  
Linear Programming Method ◽  
Low Frequencies ◽  
High Frequencies ◽  
Ar Modeling ◽  
Positive Frequency

This paper presents a method of recovering the acoustic impedance from reflection seismograms using autoregressive (AR) modeling, an approach originally applied to deconvolution by Lines and Clayton (1977). The algorithm which we describe is novel both in the manner in which the missing low‐ and high‐frequency information is predicted, and in the fact that the prediction may be constrained if acoustic impedance information is available. The prediction of the low frequencies treats the missing data as a gap which extends from the low‐frequency cut‐off in the negative frequency band to the corresponding frequency in the positive frequency band. The conjugate symmetry which governs the behavior of the spectrum in the band is taken into account in the prediction. The missing high frequencies are predicted using a modified minimum entropy norm in the frequency domain. Both synthetic and field examples are presented and illustrate the robustness of the new AR algorithm under a variety of conditions. The field example also compares the results obtained using the AR algorithm with the linear programming method of Oldenburg et al (1983). The agreement in results is particularly gratifying in view of the differences in the two inversion schemes.

Optimally Sensitive and Efficient Compact Loudspeakers for Low Audio Frequencies

2007 ◽  
Vol 38 (7) ◽  
pp. 11-17
Author(s):  
Ronald M. Aarts
Keyword(s):  
Optimality Criterion ◽  
Low Frequency ◽  
Audio Signal ◽  
Design Procedure ◽  
Frequency Range ◽  
Force Factor ◽  
Low Frequencies ◽  
Limited Frequency ◽  
The Cost ◽  
Higher Sensitivity

Conventionally, the ultimate goal in loudspeaker design has been to obtain a flat frequency response over a specified frequency range. This can be achieved by carefully selecting the main loudspeaker parameters such as the enclosure volume, the cone diameter, the moving mass and the very crucial “force factor”. For loudspeakers in small cabinets the results of this design procedure appear to be quite inefficient, especially at low frequencies. This paper describes a new solution to this problem. It consists of the combination of a highly non-linear preprocessing of the audio signal and the use of a so called low-force-factor loudspeaker. This combination yields a strongly increased efficiency, at least over a limited frequency range, at the cost of a somewhat altered sound quality. An analytically tractable optimality criterion has been defined and has been verified by the design of an experimental loudspeaker. This has a much higher efficiency and a higher sensitivity than current low-frequency loudspeakers, while its cabinet can be much smaller.

scholarly journals Bioinspired Low-Frequency Material Characterisation

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
C. Hopper ◽  
S. Assous ◽  
P. B. Wilkinson ◽  
D. A. Gunn ◽  
P. D. Jackson ◽  
...  
Keyword(s):  
Sound Velocity ◽  
Medical Physics ◽  
Low Frequency ◽  
High Sensitivity ◽  
Prediction Errors ◽  
Low Frequencies ◽  
Longitudinal Sound Velocity ◽  
Velocity Prediction ◽  
Thickness Measurements

New-coded signals, transmitted by high-sensitivity broadband transducers in the 40–200 kHz range, allow subwavelength material discrimination and thickness determination of polypropylene, polyvinylchloride, and brass samples. Frequency domain spectra enable simultaneous measurement of material properties including longitudinal sound velocity and the attenuation constant as well as thickness measurements. Laboratory test measurements agree well with model results, with sound velocity prediction errors of less than 1%, and thickness discrimination of at least wavelength/15. The resolution of these measurements has only been matched in the past through methods that utilise higher frequencies. The ability to obtain the same resolution using low frequencies has many advantages, particularly when dealing with highly attenuating materials. This approach differs significantly from past biomimetic approaches where actual or simulated animal signals have been used and consequently has the potential for application in a range of fields where both improved penetration and high resolution are required, such as nondestructive testing and evaluation, geophysics, and medical physics.

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.

scholarly journals Machine-learning based data recovery and its contribution to seismic acquisition: simultaneous application of deblending, trace reconstruction and low-frequency extrapolation

Geophysics ◽  
2020 ◽  
pp. 1-62
Author(s):  
Shotaro Nakayama ◽  
Gerrit Blacquière
Keyword(s):  
Machine Learning ◽  
Incomplete Data ◽  
Field Data ◽  
Low Frequency ◽  
Cost Effective ◽  
Complete Data ◽  
Prediction Errors ◽  
Low Frequencies ◽  
Simultaneous Application ◽  
Frequency Space

Acquisition of incomplete data, i.e., blended, sparsely sampled, and narrowband data, allows for cost-effective and efficient field seismic operations. This strategy becomes technically acceptable, provided that a satisfactory recovery of the complete data, i.e., deblended, well-sampled and broadband data, is attainable. Hence, we explore a machine-learning approach that simultaneously performs suppression of blending noise, reconstruction of missing traces and extrapolation of low frequencies. We apply a deep convolutional neural network in the framework of supervised learning where we train a network using pairs of incomplete-complete datasets. Incomplete data, which are never used for training and employ different subsurface properties and acquisition scenarios, are subsequently fed into the trained network to predict complete data. We describe matrix representations indicating the contributions of different acquisition strategies to reducing the field operational effort. We also illustrate that the simultaneous implementation of source blending, sparse geometry and band limitation leads to a significant data compression where the size of the incomplete data in the frequency-space domain is much smaller than the size of the complete data. This reduction is indicative of survey cost and duration that our acquisition strategy can save. Both synthetic and field data examples demonstrate the applicability of the proposed approach. Despite the reduced amount of information available in the incomplete data, the results obtained from both numerical and field data cases clearly show that the machine-learning scheme effectively performs deblending, trace reconstruction, and low-frequency extrapolation in a simultaneous fashion. It is noteworthy that no discernible difference in prediction errors between extrapolated frequencies and preexisting frequencies is observed. The approach potentially allows seismic data to be acquired in a significantly compressed manner, while subsequently recovering data of satisfactory quality.

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.

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.

The value of very low frequencies and far offsets for seismic data in the Permian Basin: Case study on a new dense survey from the Central Basin Platform

2022 ◽  
Vol 41 (1) ◽  
pp. 34-39
Author(s):  
Vincent Durussel ◽  
Dongren Bai ◽  
Amin Baharvand Ahmadi ◽  
Scott Downie ◽  
Keith Millis
Keyword(s):  
Seismic Data ◽  
Seismic Waves ◽  
Waveform Inversion ◽  
Low Frequency ◽  
Central Basin ◽  
Permian Basin ◽  
Depth Of Penetration ◽  
Low Frequencies ◽  
Frequency Components ◽  
Band Limited

The depth of penetration and multidimensional characteristics of seismic waves make them an essential tool for subsurface exploration. However, their band-limited nature can make it difficult to integrate them with other types of ground measurements. Consequently, far offsets and very low-frequency components are key factors in maximizing the information jointly inverted from all recorded data. This explains why extending seismic bandwidth and available offsets has become a major industry focus. Although this requirement generally increases the complexity of acquisition and has an impact on its cost, improvements have been clearly and widely demonstrated on marine data. Onshore seismic data have generally followed the same trend but face different challenges, making it more difficult to maximize the benefits, especially for full-waveform inversion (FWI). This paper describes a new dense survey acquired in 2020 in the Permian Basin and aims to objectively assess the quality and benefits brought by a richer low end of the spectrum and far offsets. For this purpose, we considered several aspects, from acquisition design and field data to FWI imaging and quantitative interpretation.

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