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- Volume 55, Issue 4, 2007
Geophysical Prospecting - Volume 55, Issue 4, 2007
Volume 55, Issue 4, 2007
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Comparison of waveform inversion, part 1: conventional wavefield vs logarithmic wavefield
Authors Changsoo Shin, Sukjoon Pyun and J. Bee BednarABSTRACTThis is the first in a series of three papers focused on using variants of a logarithmic objective function approach to full waveform inversion. In this article, we investigate waveform inversion using full logarithmic principles and compare the results with the conventional least squares approach. We demonstrate theoretically that logarithmic inversion is computational similar to the conventional method in the sense that it uses exactly the same back‐propagation technology as used in least‐squares inversion. In the sense that it produces better results for each of three numerical examples, we conclude that logarithmic inversion is also more robust. We argue that a major reason for the inherent robustness is the fact that the logarithmic approach produces a natural scaling of the amplitude of the residual wavefield by the amplitude of the modelled wavefield that tends to stabilize the computations and consequently improve the final result. We claim that any superiority of the logarithmic inversion is based on the fact that it tends to be tomographic in the early stage of the inversion and more dependent on amplitude differences in the latter stages.
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Comparison of waveform inversion, part 2: phase approach
Authors J. B. Bednar, Changsoo Shin and Sukjoon PyunABSTRACTIn this paper, we take advantage of the natural separation into amplitude and phase of a logarithmic‐based approach to full‐wavefield inversion and concentrate on deriving purely kinematic approaches for both conventional and logarithmic‐based methods. We compare the resulting algorithms theoretically and empirically. To maintain consistency between this and the previous paper in this series, we continue with the same symbolism and notation and apply our new algorithms to the same three data sets. We show that both of these new techniques, although different in implementation style, share the same computational methodology. We also show that reverse‐time back‐propagation of the residuals for our new kinematic methods continues to be the basis for calculation of the steepest‐descent vector. We conclude that the logarithmic phase‐based method is more practical than its conventionally based counterpart, but, in spite of the fact that the conventional algorithm appears unstable, differences are not great.
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Comparison of waveform inversion, part 3: amplitude approach
Authors Sukjoon Pyun, Changsoo Shin and J. B. BednarABSTRACTIn the second paper of this three part series, we studied the case of conventional and logarithmic phase‐only approaches to full‐waveform inversion. Here, we concentrate on deriving amplitude‐only approaches for both conventional‐ and logarithmic‐based methods. We define two amplitude‐only objective functions by simply assuming that the phase of the modelled wavefield is equal to that of the observed wavefield. We do this for both the conventional least‐squares approach and the logarithmic approach of Shin and Min. We show that these functions can be optimized using the same reverse‐time propagation algorithm of the full conventional methodology. Although the residuals in this case are not really residual wavefields, they can both be considered and utilized in that sense. In contrast to the case for our phase‐only algorithms, we show through numerical tests that the conventional amplitude‐only inversion is better than the logarithmic method.
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Modelling elastic properties of impure chalk from South Arne Field, North Sea
Authors Ida L. Fabricius, Christian Høier, Peter Japsen and Uffe KorsbechABSTRACTIn impure chalk, the elastic moduli are not only controlled by porosity but also by contact‐cementation, resulting in relatively large moduli for a given porosity, and by admixtures of clay and fine silica, which results in relatively small moduli for a given porosity. Based on a concept of solids suspended in pore fluids as well as composing the rock frame, we model P‐wave and S‐wave moduli of dry and wet plug samples by an effective‐medium Hashin–Shtrikman model, using chemical, mineralogical and textural input. For a given porosity, the elastic moduli correspond to a part of the solid (the iso‐frame value) forming the frame of an Upper Hashin–Shtrikman bound, whereas the remaining solid is modelled as suspended in the pore fluid. The iso‐frame model is thus a measure of the pore‐stiffness or degree of cementation of the chalk.
The textural and mineralogical data may be assessed from logging data on spectral gamma radiation, density, sonic velocity and water saturation in a hydrocarbon zone, whereas the iso‐frame value of a chalk may be assessed from the density and acoustic P‐wave logs alone. The iso‐frame concept may thus be directly used in conventional log‐analysis and is a way of incorporating sonic‐logging data.
The Rigs‐1 and Rigs‐2 wells in the South Arne field penetrate the chalk at the same depth but differ in porosity and in water saturation although almost the entire chalk interval has irreducible water saturation. Our model, combined with petrographic data, indicates that the difference in porosity is caused by a higher degree of pore‐filling cementation in Rigs‐1. Petrographic data indicate that the difference in water saturation is caused by a higher content of smectite in the pores of Rigs‐1. In both wells, we find submicron‐size diagenetic quartz.
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Nature of the scattered seismic response from zones of random clusters of cavities and fractures in a massive rock
Authors V.B. Leviant, I.B. Petrov, F.B. Chelnokov and I.Y. AntonovaABSTRACTThe scattering of elastic energy by random clusters of fractures and/or cavities in a massive rock is studied. The interpretation of the scattered seismic response reveals crucial information about the clusters of inhomogeneities (fractures/cavities), which may correspond to reservoirs. The study is based on a new two‐dimensional numerical‐modelling method that relaxes the constraints on the location and orientation of the inhomogeneities, accounts for inhomogeneities that have almost no volume but a finite surface area (fractures) and improves the accuracy of the calculation when the size of the inhomogeneities is comparable to the mesh size.
It is shown that the nature of the seismic response of zones of diffuse fracturing and/or cavities is associated with the non‐uniformity of micro‐inhomogeneities in such zones; accumulations of these micro‐inhomogeneities are known as clusters.
The relationship between the non‐uniformity of micro‐inhomogeneities and the strength of the seismic response has been established and measured.
Considerable differences in the structure of the seismic response of zones of diffuse fracturing and diffuse cavities have been identified. Converted PS‐waves dominate in the scattered wavefield associated with fractures. This is explained, as the modelling results show, by a greater transparency of fluid‐filled fractures, which reduces the reflected energy of compressional waves. The wavefield associated with cavities is characterized by the predominance (in terms of strength) of compressional PP‐waves. The strength of converted PS‐waves in the scattered wavefields for both media is approximately the same.
On the whole, according to the results of the modelling, the energy of the scattered response of fractured reservoirs is considerably less (about two times) than that of cavernous reservoirs.
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Velocity model updating in prestack Kirchhoff time migration for PS converted waves: Part I – Theory
Authors Hengchang Dai and Xiang‐Yang LiABSTRACTA velocity model updating approach is developed based on moveout analysis of the diffraction curve of PS converted waves in prestack Kirchhoff time migration. The diffraction curve can be expressed as a product of two factors: one factor depending on the PS converted‐wave velocity only, and the other factor depending on all parameters. The velocity‐dependent factor represents the hyperbolic behaviour of the moveout and the other is a scale factor that represents the non‐hyperbolic behaviour of the moveout. This non‐hyperbolic behaviour of the moveout can be corrected in prestack Kirchhoff time migration to form an inverse normal‐moveout common‐image‐point gather in which only the hyperbolic moveout is retained. This hyperbolic moveout is the moveout that would be obtained in an isotropic equivalent medium. A hyperbolic velocity is then estimated from this gather by applying hyperbolic moveout analysis. Theoretical analysis shows that for any given initial velocity, the estimated hyperbolic velocity converges by an iterative procedure to the optimal velocity if the velocity ratio is optimal or to a value closer to the optimal velocity if the velocity ratio is not optimal. The velocity ratio (VP/VS) has little effect on the estimation of the velocity. Applying this technique to a synthetic seismic data set confirms the theoretical findings. This work provides a practical method to obtain the velocity model for prestack Kirchhoff time migration.
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Velocity model updating in prestack Kirchhoff time migration for PS converted waves: Part II – Application
Authors Hengchang Dai and Xiang‐Yang LiABSTRACTWe have developed a practical approach for updating the velocity of PS converted waves based on the inverse normal‐moveout common‐image‐point gather obtained from prestack Kirchhoff time migration. We have integrated all the steps involved in updating the migration velocity model into an interactive tool and have applied this approach to a real seismic data set from the Alba Field in the North Sea. Based on experience in handling the real data, we discuss various practical aspects of updating the velocity model, including: what kind of initial velocity model should be used; which parameters in the velocity model should be updated; and how to update them. Application of prestack Kirchhoff time migration to the data set using the updated velocity model produces an improved image of the Alba Field.
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Asymptotic analysis for shear waves in a fluid‐filled borehole
Authors Gui‐jin Yao, Ke‐xie Wang, Jun Ma, Bao‐jun Yang and Hai‐rong ZhangABSTRACTWe focus on the theoretical analysis of the resonance phenomena and the geometric attenuation behaviour of critical refracted shear waves propagating along a fluid‐filled borehole. Using integration by parts, we asymptotically expand the vertical branch‐cut integral of shear waves in an infinite series related to each order of the derivative of the response function of the formation. It is proved that the vertical branch‐cut integral of shear waves at large offsets consists mainly of the contribution of the second asymptotic series, which is related to the first derivative of the response function of the formation at the shear branch point. Using the asymptotic expansion, we develop a simplified amplitude expression for shear waves, and the resonant frequency formula. The validity of the resonance frequencies obtained by the resonant frequency formula is verified numerically by comparison with the corresponding frequencies of the numerical integral results. We also give a rational explanation for the phenomenon of two peaks appearing within each resonant peak zone: i.e. that these are the contributions of the constructive interference of the shear waves and the mode poles.
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Marquardt optimization of gravity anomalies of anticlinal and synclinal structures with prescribed depth‐dependent density
Authors V. Chakravarthi and N. SundararajanABSTRACTAn inversion technique using the Marquardt optimization is developed to interpret the gravity anomalies due to anticlinal and synclinal structures with density contrast varying continuously with depth. The algorithm simultaneously estimates the parameters of the respective models, in addition to the regional gravity background that is invariably associated with the residual gravity anomaly. Forward modelling is realized through analytically derived gravity expressions for the respective models in the space domain. The efficacy of the inversion is demonstrated with the gravity anomaly due to a theoretical model, in each case with and without the regional background. In addition, the applicability is illustrated using the gravity anomalies of the Pays De Bray anticline, situated north‐west of Paris, France. The interpreted depth of the Pays De Bray anticline using the present inversion compares well with the drilling depth.
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Bayesian inference of the Cole–Cole parameters from time‐ and frequency‐domain induced polarization
Authors A. Ghorbani, C. Camerlynck, N. Florsch, P. Cosenza and A. RevilABSTRACTThe inversion of induced‐polarization parameters is important in the characterization of the frequency electrical response of porous rocks. A Bayesian approach is developed to invert these parameters assuming the electrical response is described by a Cole–Cole model in the time or frequency domain. We show that the Bayesian approach provides a better analysis of the uncertainty associated with the parameters of the Cole–Cole model compared with more conventional methods based on the minimization of a cost function using the least‐squares criterion. This is due to the strong non‐linearity of the inverse problem and non‐uniqueness of the solution in the time domain. The Bayesian approach consists of propagating the information provided by the measurements through the model and combining this information with a priori knowledge of the data. Our analysis demonstrates that the uncertainty in estimating the Cole–Cole model parameters from induced‐polarization data is much higher for measurements performed in the time domain than in the frequency domain. Our conclusion is that it is very difficult, if not impossible, to retrieve the correct value of the Cole–Cole parameters from time‐domain induced‐polarization data using standard least‐squares methods. In contrast, the Cole–Cole parameters can be more correctly inverted in the frequency domain. These results are also valid for other models describing the induced‐polarization spectral response, such as the Cole–Davidson or power law models.
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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 35 (1987)
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Volume 34 (1986)
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Volume 30 (1982)
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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 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)