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- Volume 33, Issue 6, 1985
Geophysical Prospecting - Volume 33, Issue 6, 1985
Volume 33, Issue 6, 1985
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IDENTIFICATION OF SEISMIC REFLECTIONS USING SINGULAR VALUE DECOMPOSITION*
More LessAbstractSingular value decomposition (SVD) is applied to the identification of seismic reflections by using two different models: the impulse response model, where a seismic trace is assumed to consist of a known signal pulse convolved with a reflection coefficient series plus noise, and the delayed pulse model, where the seismic signal is assumed to consist of a small number of delayed pulses of known shape and with unknown amplitudes and arrival times.
SVD clearly shows how least‐squares estimation of the reflection coefficients may become unstable, since a division by the singular values is required. Two methods for stabilizing this procedure are investigated. The inverse of the singular values may be replaced by zeros when they are less than a given threshold. This is called the SVD cut‐off method. Alternatively, we may use ridge regression which in filter design corresponds to assuming white noise. Statistical methods are used to compute an optimal SVD cut‐off level and also to compute an optimal weighting parameter in ridge regression. Numerical studies indicate that the use of SVD cut‐off or ridge regression stabilizes the least‐squares procedure, but that the results are inferior to maximum‐likelihood estimation where the noise is assumed to be filtered white noise.
For the delayed pulse model, we use a linearization procedure to iteratively update the estimates of both the reflection amplitudes and the arrival times. In each step, the optimal SVD cut‐off method is used. Confidence regions for the estimated reflection amplitudes and arrival times are also computed. Synthetic data examples demonstrate the effectiveness of this method. In a real data example, the maximum‐likelihood method assuming an impulse response model is first used to obtain initial estimates of the number of reflections and their amplitudes and traveltimes. Then the iterative procedure is used to obtain improved estimates of the reflection amplitudes and traveltimes.
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CALCULATION OF DISPERSION CURVES AND AMPLITUDE‐DEPTH DISTRIBUTIONS OF LOVE CHANNEL WAVES IN HORIZONTALLY‐LAYERED MEDIA*
More LessAbstractWe present dispersion curves, and amplitude‐depth distributions of the fundamental and first higher mode of Love seam waves for two characteristic seam models. The first model consists of four layers, representing a coal seam underlain by a root clay of variable thickness. The second model consists of five layers, representing coal seams containing a dirt band with variable position and thickness. The simple three‐layer model is used for reference.
It is shown that at higher frequencies, depending on the thickness of the root clay and the dirt band, the coal layers alone act as a wave guide, whereas at low frequencies all layers act together as a channel. Depending on the thickness, and position of the dirt band and the root clay, in the dispersion curves of the group velocity, secondary minima grow in addition to the absolute minima. Furthermore, the dispersion curves of the group velocity of the two modes can overlap. In all these cases, wave groups in addition to the Airy phase of the fundamental mode (propagating with minimum group velocity) occur on the seismograms recorded in in‐seam seismic surveys, thus impeding their interpretation. Hence, we suggest the estimation of the dispersion characteristics of Love seam waves in coal seams under investigation preceding actual field surveys.
All numerical calculations were performed using a fast and stable phase recursion algorithm.
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INSUFFICIENTLY SAMPLED Q: A CAUSE OF WAVELET DISTORTION IN THE GENERATION OF SYNTHETIC SEISMOGRAMS?*
By K. MOSEGAARDAbstractThe effect of sampling of Q‐values on the generation of one dimensional synthetic seismograms including absorption and dispersion is investigated. For the well data considered, a decrease in the Q‐sampling interval results in an elongation of the estimated reflection waveform.
Two explanations are given. The first refers to the way multiple energy is absorbed in the stratified medium. The second is based on the general observation that geological media with high velocity (e.g. limestones) often have high Q‐values, whereas media with low velocity (e.g. shales) often have low Q‐values.
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MAXIMUM LIKELIHOOD ESTIMATION OF SEISMIC REFLECTION COEFFICIENTS*
By H. ÖZDEMİRAbstractA seismic trace is modeled as a moving average (MA) process both in signal and noise: a signal wavelet convolved with a reflection coefficient series plus colored random noise. Seismic reflection coefficients can be estimated from seismic traces using suitable estimation algorithms if the input wavelet is known and vice versa. The maximum likelihood (ML) algorithm is used to estimate the system order and the reflection coefficients. The system order is related to the arrival time of the latest signal in a complex seismic reflection event. The least‐squares (LS) method does not provide such information. The ML algorithm makes assumptions only about the Gaussian nature of the noise. It is better suited for seismic applications since the LS method inherits the white noise assumption. The Gauss‐Newton (G‐N) and Newton‐Raphson (N‐R) optimization algorithms are used to obtain the ML and the LS estimates.
Reflection coefficient estimations are affected by the choice of sampling rate of seismic data. Theoretically, the optimum choice in system identification is the Nyquist rate. Experience with synthetic data confirms the theory.
In practice, good estimates of reflection coefficients are possible only up to certain pulse separations (or, equivalently, orders). This is mostly due to numerical problems with the optimization algorithms used and partly due to the limited bandwidth of seismic signals. Good estimates from data simulated using three airgun array pulses recorded with 6–128 Hz filter setting are possible up to about 40.0 ms pulse separations. Successful estimations from pinchout and thin layer simulations and well controlled offshore “bright‐spots” are given.
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SIMPLE DATA PROCESSING OF TRIPOTENTIAL APPARENT RESISTIVITY MEASUREMENTS AS AN AID TO THE INTERPRETATION OF SUBSURFACE STRUCTURE*
Authors R. I. ACWORTH and D. H. GRIFFITHSAbstractThe advantages of the Wenner tripotential method (Carpenter 1955) for apparent resistivity profiling are described and two new data processing techniques introduced as an aid to the interpretation of apparent resistivity sections (pseudo‐sections). These techniques were developed from model data computed using a two‐dimensional finite difference method.
Oscillatory components present in anomalies on tripotential profiles and related to electrode spacing are shown to be effectively removed by linear filtering that also simplifies their form and aids recognition. Furthermore, it is shown that the ratio of the beta‐ and gamma‐apparent resistivities is a good indicator of resistivity variation, and is particularly sensitive to lateral change.
Model data indicates that, over a wide range of conditions, enough subsurface information can be obtained by inspection of tripotential resistivity and ratio profiles, and from space sections to make possible a useful—and sometimes semi‐quantitative—interpretation.
A rationale for the general interpretation of tripotential data is developed. Field data are described from an area of weathered granite basement in Nigeria. A model of the subsurface is developed using parameters derived from the processed observations. The observed and calculated apparent resistivity space sections are very similar.
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Volumes & issues
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Volume 72 (2023 - 2024)
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Volume 71 (2022 - 2023)
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Volume 70 (2021 - 2022)
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Volume 69 (2021)
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Volume 68 (2020)
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Volume 67 (2019)
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Volume 66 (2018)
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Volume 65 (2017)
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Volume 64 (2015 - 2016)
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Volume 63 (2015)
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Volume 62 (2014)
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Volume 61 (2013)
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Volume 60 (2012)
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Volume 59 (2011)
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Volume 58 (2010)
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Volume 57 (2009)
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Volume 56 (2008)
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Volume 55 (2007)
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Volume 54 (2006)
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Volume 53 (2005)
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Volume 52 (2004)
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Volume 51 (2003)
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Volume 50 (2002)
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Volume 49 (2001)
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Volume 48 (2000)
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Volume 47 (1999)
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Volume 46 (1998)
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Volume 45 (1997)
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Volume 44 (1996)
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Volume 43 (1995)
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Volume 42 (1994)
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Volume 41 (1993)
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Volume 40 (1992)
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Volume 39 (1991)
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Volume 38 (1990)
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Volume 37 (1989)
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Volume 36 (1988)
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Volume 35 (1987)
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Volume 34 (1986)
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Volume 33 (1985)
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Volume 32 (1984)
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Volume 31 (1983)
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Volume 30 (1982)
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Volume 29 (1981)
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Volume 28 (1980)
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Volume 27 (1979)
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Volume 26 (1978)
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Volume 25 (1977)
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Volume 24 (1976)
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Volume 23 (1975)
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Volume 22 (1974)
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Volume 21 (1973)
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Volume 20 (1972)
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Volume 19 (1971)
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Volume 18 (1970)
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Volume 17 (1969)
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Volume 16 (1968)
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Volume 15 (1967)
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Volume 14 (1966)
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Volume 13 (1965)
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Volume 12 (1964)
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Volume 11 (1963)
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Volume 10 (1962)
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Volume 9 (1961)
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Volume 8 (1960)
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Volume 7 (1959)
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Volume 6 (1958)
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Volume 5 (1957)
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Volume 4 (1956)
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Volume 3 (1955)
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Volume 2 (1954)
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Volume 1 (1953)