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- Volume 61, Issue 1, 2013
Geophysical Prospecting - Volume 61, Issue 1, 2013
Volume 61, Issue 1, 2013
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Review paper: What Kelvin might have written about Elasticity
By Klaus HelbigABSTRACTOne hundred and fifty five years ago, Kelvin published the first part of a fundamental analysis of the elastic tensor, in which he proposed a coordinate‐free representation through its eigensystem. His thoughts were apparently far ahead of his time, since it took 125 years before the paper elicited a positive reaction (it is now accessible through several modern reviews). Science not only lost track for 125 years of the original paper but also lost the ideas Kelvin might have proposed in the second part, a publication that was never put to paper, presumably in view of the lack of appreciation of the first part.
In an attempt to establish what might have been on Kelvin's mind for a second part, one has to ‘forget’ the progress of mathematical physics in the intervening time and base all arguments strictly on the content of the first part and on the state of science in the second half of the 19thcentury.
The theory of elasticity would certainly have developed faster, had Kelvin's paper peen appreciated by his ‘peers’. But a theory based on Kelvin's ideas would be fruitful even today.
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Non‐hyperbolic common reflection surface
Authors Sergey Fomel and Roman KazinnikABSTRACTThe method of common reflection surface (CRS) extends conventional stacking of seismic traces over offset to multidimensional stacking over offset‐midpoint surfaces. We propose a new form of the stacking surface, derived from the analytical solution for reflection traveltime from a hyperbolic reflector. Both analytical comparisons and numerical tests show that the new approximation can be significantly more accurate than the conventional CRS approximation at large offsets or at large midpoint separations while using essentially the same parameters.
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Comparison of sparse‐grid geometric and random sampling methods in nonlinear inverse solution uncertainty estimation
More LessABSTRACTA new uncertainty estimation method, which we recently introduced in the literature, allows for the comprehensive search of model posterior space while maintaining a high degree of computational efficiency. The method starts with an optimal solution to an inverse problem, performs a parameter reduction step and then searches the resulting feasible model space using prior parameter bounds and sparse‐grid polynomial interpolation methods. After misfit rejection, the resulting model ensemble represents the equivalent model space and can be used to estimate inverse solution uncertainty. While parameter reduction introduces a posterior bias, it also allows for scaling this method to higher dimensional problems. The use of Smolyak sparse‐grid interpolation also dramatically increases sampling efficiency for large stochastic dimensions. Unlike Bayesian inference, which treats the posterior sampling problem as a random process, this geometric sampling method exploits the structure and smoothness in posterior distributions by solving a polynomial interpolation problem and then resampling from the resulting interpolant. The two questions we address in this paper are 1) whether our results are generally compatible with established Bayesian inference methods and 2) how does our method compare in terms of posterior sampling efficiency. We accomplish this by comparing our method for two electromagnetic problems from the literature with two commonly used Bayesian sampling schemes: Gibbs’ and Metropolis‐Hastings. While both the sparse‐grid and Bayesian samplers produce compatible results, in both examples, the sparse‐grid approach has a much higher sampling efficiency, requiring an order of magnitude fewer samples, suggesting that sparse‐grid methods can significantly improve the tractability of inference solutions for problems in high dimensions or with more costly forward physics.
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Dip‐adaptive singular‐value decomposition filtering for seismic reflection enhancement
Authors Milton J. Porsani, Bjorn Ursin, Michelângelo G. Silva and Paulo E. M. MeloABSTRACTWe present a singular value decomposition (SVD) filtering method for the enhancement of coherent reflections and for attenuation of noise. The method is applied in two steps. First normal move‐out (NMO) correction is applied to shot or CMP records, with the purpose of flattening the reflections. We use a spatial SVD filter with a short sliding window to enhance coherent horizontal events. Then the data are sorted in common‐offset panels and the local dip is estimated for each panel. The next SVD filtering is performed on a small number of traces and a small number of time samples centred around the output sample position. Data in a local window are corrected for linear moveout corresponding to the dips before SVD. At the central time sample position, we sum over the dominant eigenimages of a few traces, corresponding to SVD dip filtering. We illustrate the method using land seismic data from the Tacutu basin, located in the north‐east of Brazil. The results show that the proposed method is effective and is able to reveal reflections masked by ground‐roll and other types of noise.
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Analytical wavefront curvature correction to plane‐wave reflection coefficients for a weak‐contrast interface
Authors Badr Alulaiw and Boris GurevichABSTRACTMost amplitude versus offset (AVO) analysis and inversion techniques are based on the Zoeppritz equations for plane‐wave reflection coefficients or their approximations. Real seismic surveys use localized sources that produce spherical waves, rather than plane waves. In the far‐field, the AVO response for a spherical wave reflected from a plane interface can be well approximated by a plane‐wave response. However this approximation breaks down in the vicinity of the critical angle. Conventional AVO analysis ignores this problem and always utilizes the plane‐wave response. This approach is sufficiently accurate as long as the angles of incidence are much smaller than the critical angle. Such moderate angles are more than sufficient for the standard estimation of the AVO intercept and gradient. However, when independent estimation of the formation density is required, it may be important to use large incidence angles close to the critical angle, where spherical wave effects become important. For the amplitude of a spherical wave reflected from a plane fluid‐fluid interface, an analytical approximation is known, which provides a correction to the plane‐wave reflection coefficients for all angles. For the amplitude of a spherical wave reflected from a solid/solid interface, we propose a formula that combines this analytical approximation with the linearized plane‐wave AVO equation. The proposed approximation shows reasonable agreement with numerical simulations for a range of frequencies. Using this solution, we constructed a two‐layer three‐parameter least‐squares inversion algorithm. Application of this algorithm to synthetic data for a single plane interface shows an improvement compared to the use of plane‐wave reflection coefficients.
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Dreamlet source‐receiver survey sinking prestack depth migration
Authors Bangyu Wu, Ru‐shan Wu and Jinghuai GaoABSTRACTSurvey sinking migration downward continues the entire surface observed multi‐shot data to the subsurface step by step recursively. Reflected energy from reflectors at current depth appear at zero time and zero offset in the extrapolated wavefield. The data (seismic records) of t > 0 at this depth are equivalent to the data acquired by a survey system deployed at this depth. This is the reason to name the process ‘survey sinking’. The records of negative time need not to be further propagated since they carry no information to image structures beneath the new survey system. In this paper, we combine survey sinking with dreamlet migration. The dreamlet migration method decomposes the seismic wavefield and one‐way wave propagator by complete time‐space localized bases. The localization on time gives flexibility on time‐varying operations during depth extrapolation. In dreamlet survey sinking migration, it only keeps the data for imaging the structures beneath the sunk survey system and gets rid of the data already used to image structures above it. The deeper the depth is, the shorter is the valid time records of the remaining data and less computation is needed for one depth step continuation. For data decomposition, in addition to time axis, dreamlet survey sinking also decomposes the data for source and receiver gathers, which is a fully localized decomposition of prestack seismic data. A three‐scatter model is first used to demonstrate the computational feature and principle of this method. Tests on the two‐dimensional SEG/EAGE salt model show that with reduced data sets the proposed method can still obtain good imaging quality on complex geology structures and a strong velocity contrast environment.
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Wide‐azimuth angle gathers for anisotropic wave‐equation migration
Authors Paul Sava and Tariq AlkhalifahABSTRACTExtended common‐image‐point gathers (CIP) constructed by wide‐azimuth TI wave‐equation migration contain all the necessary information for angle decomposition as a function of the reflection and azimuth angles at selected locations in the subsurface. The aperture and azimuth angles are derived from the extended images using analytic relations between the space‐ and time‐lag extensions using information which is already available at the time of migration, i.e. the anisotropic model parameters. CIPs are cheap to compute because they can be distributed in the image at the most relevant positions, as indicated by the geologic structure. If the reflector dip is known at the CIP locations, then the computational cost can be reduced by evaluating only two components of the space‐lag vector. The transformation from extended images to angle gathers is a planar Radon transform which depends on the local medium parameters. This transformation allows us to separate all illumination directions for a given experiment, or between different experiments. We do not need to decompose the reconstructed wavefields or to choose the most energetic directions for decomposition. Applications of the method include illumination studies in complex areas where ray‐based methods fail, and assuming that the subsurface illumination is sufficiently dense, the study of amplitude variation with aperture and azimuth angles.
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Short‐period static correction without Normal Move Out†
Authors Andrey V. Masjukov and Igor S. MurzovABSTRACTStatic correction is a common step in a seismic data proccessing flowchart for land data. Here we propose a new algorithm for automatic short‐period static correction. The algorithm is based on the assumption that seismic events after short‐period static correction should be locally plane nearly everywhere. No other assumptions are made. Therefore the proposed method does not require a preliminary velocity analysis. The algorithm consists in two main parts: evaluation of second spatial differences of trajectories and subsequent regularized integration of these differences. The proposed method proves its robustness and shows results comparable with conventional residual static correction based on improving common‐midpoint stacking. In contrast to the conventional residual static, the proposed algorithm can estimate short‐period statics in complex cases where common‐midpoint stacking fails because of non‐hyperbolic events.
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The dynamic response of a fluid‐filled borehole under a normal point load on the free surface
More LessABSTRACTThe dynamic response of a semi‐infinite fluid‐filled borehole embedded in an elastic half‐space under a concentrated normal surface load is analysed in the long‐wavelength limit. The solution of the problem is obtained with integral transforms in the form of a double integral with respect to the slowness and frequency. The partial P‐ and SV‐wave responses are further transformed to path integrals along Cagniard paths in the complex slowness plane. Unlike the traditional Cagniard‐de Hoop technique based on the Laplace transform of time dependence, this paper is based on the Fourier transform. The tube‐wave response is presented as a causal integral over a slowness range. The resultant representation in the time‐domain is suitable for the numerical evaluation of the complete response in the fluid‐filled borehole, especially at large distances.
Asymptotic analysis of seismic phases arising in the borehole is performed on the basis of the obtained solution. The complete asymptotic wavefield consists in P‐ and SV‐waves, the Rayleigh wave and the low‐frequency Stoneley (tube) wave. Pressure synthetics obtained by the use of the asymptotic formulas are shown to be in good agreement with straightforward calculations.
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Permeability inversion from low‐frequency seismoelectric logs in fluid‐saturated porous formations
Authors Wei Guan, Hengshan Hu and Zhi WangABSTRACTIn this paper, a method is proposed to invert permeability from seismoelectric logs in fluid‐saturated porous formations. From the analysis of both the amplitude and the phase of simulated seismoelectric logs, we find that the Stoneley wave amplitude of the ratio of the converted electric field to the pressure (REP) is sensitive to porosity rather than permeability while the tangent of the REP's phase is sensitive to permeability. The REP's phase reflects the phase discrepancy between the electric field and the pressure at the same location in the borehole. We theoretically derive the frequency‐dependent expression of the REP of the low‐frequency Stoneley wave and find that the tangent of the REP's argument is approximately in inverse proportion to permeability. We obtain an inversion formula and present the permeability inversion method by using the tangent of the REP's phase. To test this method, the permeabilities of different sandstones are inverted from the synthetic full‐waveform data of seismoelectric logs. A modified inversion process is proposed based on the analysis of inversion errors, by which the relative errors are controlled below 25% and they are smaller than those of the permeability inversion from the Stoneley wave of acoustic logs.
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Localization and characterization of cracks in clay‐rocks using frequency and time‐domain induced polarization
Authors G. Okay, P. Cosenza, A. Ghorbani, C. Camerlynck, J. Cabrera, N. Florsch and A. RevilABSTRACTInduced polarization (IP) is a geophysical method that is potentially sensitive to the presence of cracks in porous rocks and therefore to damage. We performed time‐domain and frequency domain IP measurements at the Tournemire Underground Research Laboratory (URL, Aveyron, France) in areas where different types of cracks are observed. These cracks correspond to both tectonic fractures and new cracks associated with stress release and desiccation resulting from the excavation of a gallery. These measurements were performed both in eastern and northern galleries of the test site. The eastern gallery was excavated in 1996 while the northern gallery was excavated recently in 2008. This gives us the opportunity to study the electrical characteristics of the excavation damaged zone surrounding the galleries with respect to the age of the excavation. Longitudinal profiles were performed along the floor of the galleries with 48 Cu/CuSO4 electrodes separated by a distance of 20 cm. Chargeability and resistivity were inverted using a Gauss‐Newton iterative approach assuming an isotropic heterogeneous clay‐rock material. The resulting IP tomograms show a correlation between high values of chargeability and the presence of calcite‐filled tectonic fractures. X‐ray analysis indicates that the presence of pyrite in these fractures is a potential source of the observed IP signals. The cracks associated with the mechanical damage of the formation exhibit low values of chargeability, on the same order of magnitude than the chargeability of the clay‐rock matrix and are therefore hardly observable. A smaller IP response associated with the presence of these cracks is observed in the older gallery and this observation is qualitatively related to the desaturation process associated with these cracks. In a specific area of one of the galleries, the presence of calcareous nodules is observed to be an important source of anomalous chargeability. This signature seems to be associated with the presence of pyrite.
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Experimental estimation of velocities and anisotropy of a series of Swedish crystalline rocks and ores
ABSTRACTTo provide a guide for future deep (<1.5 km) seismic mineral exploration and to better understand the nature of reflections imaged by surface reflection seismic data in two mining camps and a carbonatite complex of Sweden, more than 50 rock and ore samples were collected and measured for their seismic velocities. The samples are geographically from the northern and central parts of Sweden, ranging from metallic ore deposits, meta‐volcanic and meta‐intrusive rocks to deformed and metamorphosed rocks. First, ultrasonic measurements of P‐ and S‐wave velocities at both atmospheric and elevated pressures, using 0.5 MHz P‐ and S‐wave transducers were conducted. The ultrasonic measurements suggest that most of the measured velocities show positive correlation with the density of the samples with an exception of a massive sulphide ore sample that shows significant low P‐ and S‐wave velocities. The low P‐ and S‐wave velocities are attributed to the mineral texture of the sample and partly lower pyrite content in comparison with a similar type sample obtained from Norway, which shows significantly higher P‐ and S‐wave velocities. Later, an iron ore sample from the central part of Sweden was measured using a low‐frequency (0.1–50 Hz) apparatus to provide comparison with the ultrasonic velocity measurements. The low‐frequency measurements indicate that the iron ore sample has minimal dispersion and attenuation. The iron ore sample shows the highest acoustic impedance among our samples suggesting that these deposits are favourable targets for seismic methods. This is further demonstrated by a real seismic section acquired over an iron ore mine in the central part of Sweden. Finally, a laser‐interferometer device was used to analyse elastic anisotropy of five rock samples taken from a major deformation zone in order to provide insights into the nature of reflections observed from the deformation zone. Up to 10% velocity‐anisotropy is estimated and demonstrated to be present for the samples taken from the deformation zone using the laser‐interferometery measurements. However, the origin of the reflections from the major deformation zone is attributed to a combination of anisotropy and amphibolite lenses within the deformation zone.
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P‐wave velocity distribution in basalt flows of the Enni Formation in the Faroe Islands from refraction seismic analysis
Authors Uni K. Petersen, R. James Brown and Morten S. AndersenABSTRACTThe main objective of this work is to establish the applicability of shallow surface‐seismic traveltime tomography in basalt‐covered areas. A densely sampled ∼1300‐m long surface seismic profile, acquired as part of the SeiFaBa project in 2003 (Japsen et al. 2006) at Glyvursnes in the Faroe Islands, served as the basis to evaluate the performance of the tomographic method in basalt‐covered areas. The profile is centred at a ∼700‐m deep well. VP, VS and density logs, a zero‐offset VSP, downhole‐geophone recordings and geological mapping in the area provided good means of control.
The inversion was performed with facilities of the Wide Angle Reflection/Refraction Profiling program package (Ditmar et al. 1999). We tested many inversion sequences while varying the inversion parameters. Modelled traveltimes were verified by full‐waveform modelling.
Typically an inversion sequence consists in several iterations that proceed until a satisfactory solution is reached. However, in the present case with high velocity contrasts in the subsurface we obtained the best result with two iterations: first obtaining a smooth starting model with small traveltime residuals by inverting with a high smoothing constraint and then inverting with the lowest possible smoothing constraint to allow the inversion to have the full benefit of the traveltime residuals.
The tomogram gives usable velocity information for the near‐surface geology in the area but fails to reproduce the expected velocity distribution of the layered basalt flows. Based on the analysis of the tomogram and geological mapping in the area, a model was defined that correctly models first arrivals from both surface seismic data and downhole‐geophone data.
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A numerical comparison of time and frequency‐domain marine electromagnetic methods for hydrocarbon exploration in shallow water
Authors Dylan Connell and Kerry KeyABSTRACTIn shallow water the frequency domain controlled source electromagnetic method is subject to airwave saturation that strongly limits the sensitivity to resistive hydrocarbon targets at depth. It has been suggested that time‐domain CSEM may offer an improved sensitivity and resolution of these deep targets in the presence of the airwave. In order to examine and test these claims, this work presents a side‐by‐side investigation of both methods with a main focus on practical considerations, and how these effect the resolution of a hydrocarbon reservoir. Synthetic noisy data for both time‐domain and frequency domain methods are simulated using a realistic frequency dependent noise model and frequency dependent scaling for representative source waveforms. The synthetic data studied here include the frequency domain response from a compact broadband waveform, the time‐domain step‐response from a low‐frequency square wave and the time‐domain impulse response obtained from pseudo‐random binary sequences. These data are used in a systematic resolution study of each method as a function of water‐depth, relative noise and stacking length. The results indicate that the broadband frequency domain data have the best resolution for a given stacking time, whereas the time‐domain data require prohibitively longer stacking times to achieve similar resolution.
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Imaging the Kristineberg mining area with two perpendicular magnetotelluric profiles in the Skellefte Ore District, northern Sweden
More LessABSTRACTNew magnetotelluric (MT) data from two perpendicular profiles in the Kristineberg area, northern Sweden, were analysed and modelled. In the Skellefte Ore District, the Kristineberg volcanic hosted massive sulphide (VHMS) deposit mine is one of the largest and deepest (1250 m). Seventeen broadband magnetotelluric stations were installed along two existing seismic reflection lines. The profiles were 6 and 12 km long with 500 m and 1 km site spacing, respectively. The obtained MT transfer functions in the period range of 0.0015–200 s are of fairly good quality. Detailed strike and dimensionality analysis reveal consistent but period dependent, strike directions, indicating a change in the geoelectrical strike with depth. From the two‐dimensional inversion of the determinant of the impedance tensor, two stable conductivity models with good data fit were obtained. The addition of seismic reflection information from the co‐located survey, improved the data fit of one of them. Extensive sensitivity analyses helped to delineate the well resolved regions of the models and to determine the position of pronounced boundaries. The results are in good agreement with previous studies, especially regarding the presence of a deep conductor interpreted as a structural basement to the district. They also reveal with more detail the configuration of the main geological units of the Skellefte Ore District, especially of the ore bearing volcanic rocks and the embedded alteration zones.
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Gravity inversion of 2D bedrock topography for heterogeneous sedimentary basins based on line integral and maximum difference reduction methods
More LessABSTRACTBased on the line integral (LI) and maximum difference reduction (MDR) methods, an automated iterative forward modelling scheme (LI‐MDR algorithm) is developed for the inversion of 2D bedrock topography from a gravity anomaly profile for heterogeneous sedimentary basins. The unknown basin topography can be smooth as for intracratonic basins or discontinuous as for rift and strike‐slip basins. In case studies using synthetic data, the new algorithm can invert the sedimentary basins bedrock depth within a mean accuracy better than 5% when the gravity anomaly data have an accuracy of better than 0.5 mGal. The main characteristics of the inversion algorithm include: (1) the density contrast of sedimentary basins can be constant or vary horizontally and/or vertically in a very broad but a priori known manner; (2) three inputs are required: the measured gravity anomaly, accuracy level and the density contrast function, (3) the simplification that each gravity station has only one bedrock depth leads to an approach to perform rapid inversions using the forward modelling calculated by LI. The inversion process stops when the residual anomalies (the observed minus the calculated) falls within an ‘error envelope’ whose amplitude is the input accuracy level. The inversion algorithm offers in many cases the possibility of performing an agile 2D gravity inversion on basins with heterogeneous sediments. Both smooth and discontinuous bedrock topography with steep spatial gradients can be well recovered. Limitations include: (1) for each station position, there is only one corresponding point vertically down at the basement; and (2) the largest error in inverting bedrock topography occurs at the deepest points.
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Effect of sub‐core scale heterogeneities on acoustic and electrical properties of a reservoir rock: a CO2 flooding experiment of brine saturated sandstone in a computed tomography scanner
Authors Binyam L. Alemu, Eyvind Aker, Magnus Soldal, Øistein Johnsen and Per AagaardABSTRACTThe effect of sub‐core scale heterogeneity on fluid distribution pattern, and the electrical and acoustic properties of a typical reservoir rock was studied by performing drainage and imbibition flooding tests with CO2 and brine in a laboratory. Moderately layered Rothbach sandstone was used as a test specimen. Two core samples were drilled; one perpendicular and the other parallel to the layering to allow injection of fluids along and normal to the bedding plane. During the test 3D images of fluid distribution and saturation levels were mapped by an industrial X‐ray CT‐scanner together with simultaneous measurement of electrical resistivity, ultrasonic velocities as well as amplitudes.
The results showed how the layering and the flooding direction influenced the fluid distribution pattern and the saturation level of the fluids. For a given fluid saturation level, the measured changes in the acoustic and electrical parameters were affected by both the fluid distribution pattern and the layering orientation relative to the measurement direction. The P‐wave amplitude and the electrical resistivity were more sensitive to small changes in the fluid distribution patterns than the P‐wave velocity. The change in amplitude was the most affected by the orientation of the layering and the resulting fluid distribution patterns. In some instances the change due to the fluid distribution pattern was higher than the variation caused by the change in CO2 saturation. As a result the Gassmann relation based on ‘uniform' or ‘patchy' saturation pattern was not suitable to predict the P‐wave velocity variation. Overall, the results demonstrate the importance of core‐imaging to improve our understanding of fluid distribution patterns and the associated effects on measured rock‐physics properties.
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