Bjørn Ursin

Focusing in dip and AVA compensation on scattering‐angle/azimuth common image gathers

Common image gathers (CIGs) in the offset and surface azimuth domain are used extensively in migration velocity analysis and amplitude variation with offset (AVO) studies. If the geology is complex and the ray field becomes multipathed, the quality of the CIGs deteriorates. To overcome these problems, the CIGs are generated as a function of scattering angle and azimuth at the image point. The CIGs are generated using an algorithm based on the inverse generalized Radon transform (GRT), stacking only over migration dip angles. Including only dips in the vicinity of the geological dip, or focusing in dip, suppresses artifacts in and results in improved signal‐to‐noise ratio on the CIGs.Migration velocity analysis can be based upon the differential semblance criterion. The analysis~is~then carried out by minimizing a functional of the derivative of the CIGs with respect to horizontal coordinates (offset/azimuth or scattering‐angle/azimuth), but AVO/amplitude variation with angle (AVA) effects will degrade the...

Decomposition of electromagnetic fields into upgoing and downgoing components

This paper gives a unified treatment of electromagnetic EM field decomposition into upgoing and downgoing components forconductiveandnonconductivemedia,wheretheelectromagneticdataaremeasuredonaplaneinwhichtheelectricpermittivity,magneticpermeability,andelectricalconductivityareknown constants with respect to space and time. Above and below the plane of measurement, the medium can be arbitrarily inhomogeneousandanisotropic. In particular, the proposed decomposition theory applies to marine EM, low-frequency data acquired for hydrocarbon mapping where the upgoing components of the recordedfield guided and refracted from the reservoir, that are of interest for the interpretation. The direct-source field, the refracted airwave induced by the source, the reflected field from the sea surface, and most magnetotelluric noise traveling downward just below the seabed are field components that are considered to be noise in electromagneticmeasurements. The viability and validity of the decomposition method is demonstrated using modeled and real marine EM data, also termed seabed logging SBL data. The synthetic data are simulated in a model that is fairly representative of the geologic area wheretherealSBLwerecollected.Theresultsfromthesynthetic data study therefore are used to assist in the interpretation of the realdatafromanareawith320-mwaterdepthaboveaknowngas province offshore Norway. The effect of the airwave is seen clearly in measured data. After field decomposition just below the seabed, the upgoing component of the recorded electric field has almost linear phase, indicating that most of the effect of the airwavecomponenthasbeenremoved.

Traveltime approximations for a layered transversely isotropic medium

We consider multiple transmitted, reflected, and converted qP-qSV-waves or multiple transmitted and reflected SH-waves in a horizontally layered medium that is transversely isotropic with a vertical symmetry axis (VTI). Traveltime and offset (horizontal distance) between a source and receiver, not necessarily in the same layer, are expressed as functions of horizontal slowness. These functions are given in terms of a Taylor series in slowness in exactly the same form as for a layered isotropic medium. The coefficients depend on the parameters of the anisotropic layers through which the wave has passed, and there is no weak anisotropy assumption. Using classical formulas, the traveltime or traveltime squared can then be expressed as a Taylor series in even powers of offset. These Taylor series give rise to a shifted hyperbola traveltime approximation and a new continued-fraction approximation, described by four parameters that match the Taylor series up to the sixth power in offset. Further approximations give several simplified continued-fraction approximations, all of which depend on three parameters: zero-offset traveltime, NMO velocity, and a heterogeneity coefficient. The approximations break down when there is a cusp in the group velocity for the qSV-wave. Numerical studies indicate that approximations of traveltime squared are generally better than those for traveltime. A new continued-fraction approximation that depends on three parameters is more accurate than the commonly used continued-fraction approximation and the shifted hyperbola.

Cross-property relations between electrical conductivity and the seismic velocity of rocks

Cross-property relations are useful when some rock properties can be measured more easily than other properties. Relations between electrical conductivity and seismic velocity, stiffness moduli, and density can be obtained by expressing the porosity in terms of those properties. There are many possible ways to combine the constitutive equations to obtain a relation, each one representing a given type of rock. The relations depend on the assumptions to obtain the constitutive equations. In the electromagnetic case, the equations involve Archie’s law and its modifications for a conducting frame, the Hashin-Shtrikman HS bounds, and the self-similar and complex refraction-index method CRIM models. In the elastic case, the stress-strain relations are mainly based on the time-average equation, the HS bounds, and the Gassmann equation. Also, expressions for dry rocks and for anisotropic media, using Backus averaging, are analyzed. The relations are applied to a shale saturated with brineoverburden and to a sandstone saturated with oilreservoir. Tests with sections of a North Sea well log show that the best fit is given by the relation between the Gassmann velocity and the CRIM, selfsimilar, and Archie models for the conductivity.

Offset‐dependent geometrical spreading in a layered medium

The geometrical spreading for a point source in a horizontally layered medium has been computed by Ursin (1978) and Hubral (1978) as a Taylor series in the offset coordinate. The coefficients in the Taylor series depend on the thicknesses and the velocities of the layers. Here, I start with the exact expression for geometrical spreading and show that it can be expressed as a function of the velocity in the first layer, the offset, and the first‐ and second‐order traveltime derivatives. A shifted hyperbolic traveltime approximation (Castle, 1988) and the usual hyperbolic traveltime approximation are used to derive approximate expressions for geometrical spreading. These expressions can also be derived from a truncated Taylor series by making additional approximations, but this procedure is not so obvious.

Inversion of reflection times in three dimensions

When reflection data are available from a grid of crossing seismic lines, it is possible to construct normal incidence time maps from interpreted stacked sections and then apply three‐dimensional (3-D) ray‐tracing techniques following the normal‐incidence raypaths down to the various reflectors. The main disadvantage of this well‐known “time map migration” procedure is that interval velocities must be known a priori, and they must be estimated in advance by some approximate method. A technique is presented here which combines the above procedure with an inversion algorithm, providing direct calculations of interval velocities from the additional use of nonzero offset traveltime observations. A generalized linear inversion scheme is used, making possible a complete calculation of interval velocities and reflection interfaces, the latter represented by bicubic spline functions. To test the method in practice, we have applied it to (1) synthetic data generated from a constructed model, and (2) real data obta...

Reflection and transmission responses of a layered isotropic viscoelastic medium

Transmission effects in the overburden are important for amplitude versus offset (AVO) studies and for true‐amplitude imaging of seismic data. Thin layers produce transmission effects which depend on frequency and slowness. We consider an inhomogeneous viscoelastic layered isotropic medium where the parameters depend on depth only. This takes into account both the effects of intrinsic attenuation and the effects of the layering (including changes in attenuation). The seismic wavefield is decomposed into up‐ and downgoing waves scaled with respect to the vertical energy flux. This gives important symmetry relations for the reflection and transmission responses. For a stack of homogeneous layers, the exact reflection response can be computed in a numerically stable way by a simple layer‐recursive algorithm.The reflection and transmission coefficients at a plane interface are functions of the complex medium parameters (depending on frequency) and the real horizontal slowness parameter. Approximations for wea...

Preconditioning of full-waveform inversion in viscoacoustic media

Intrinsic absorption in the earth affects the amplitude and phase spectra of the seismic wavefields and records, and may degrade significantly the results of acoustic full-waveform inversion. Amplitude distortion affects the strength of the scatterers and decreases the resolution. Phase distortion may result in mislocated interfaces. We show that viscoacoustic gradient-based inversion algorithms (e.g., steepest descent or conjugate gradients) compensate for the effects of phase distortion, but not for the effects of amplitude distortion. To solve this problem at a reasonable numerical cost, we have designed two new forms of preconditioning derived from an analysis of the inverse Hessian operator. The first type of preconditioning is a frequency-dependent compensation for dispersion and attenuation, which involves two extra modeling steps with inverse absorption (amplification) at each iteration. The second type only corrects the strength of the recovered scatterers, and requires two extra modeling steps at the first iteration only. The new preconditioning methods have been incorporated into a finite-difference inversion scheme for viscoacoustic media. Numerical tests on noise-free synthetic data illustrate and support the theory.

Low-frequency electromagnetic fields in applied geophysics: Waves or diffusion?

Low-frequency electromagnetic (EM) signal propagation in geophysical applications is sometimes referred to as diffusion and sometimes as waves. In the following we discuss the mathematical and physical approaches behind the use of the different terms. The basic theory of EM wave propagation is reviewed. From a frequency-domain description we show that all of the well-known mathematical tools of wave theory, including an asymptotic ray-series description, can be applied for both nondispersive waves in nonconductive materials and low-frequency waves in conductive materials. We consider the EM field from an electric dipole source and show that a common frequency-domain description yields both the undistorted pulses in nonconductive materials and the strongly distorted pulses in conductive materials. We also show that the diffusion-equation approximation of low-frequency EM fields in conductive materials gives the correct mathematical description, and this equation has wave solutions. Having considered both a wave-picture approach and a diffusion approach to the problem, we discuss the possible confusion that the use of these terms might lead to.

Comparison of seismic attenuation models using zero-offset vertical seismic profiling (VSP) data

For seismic frequencies it is common to use an empirical equation to model attenuation. Usually the attenuation coefficient is modeled with linear frequency dependence, a model referred to as the Kolsky-Futterman model. Other models have been suggested in the geophysical literature. We compare eight of these models on a zero-offset vertical seismic profiling (VSP) data set: the Kolsky-Futterman, the power law, the Kjartansson, the Muller, the Azimi second, the Azimi third, the Cole-Cole, and the standard linear solid (SLS) models. For three separate depth zones we estimate velocities and Q-values for all eight models. A least-squares model-fitting algorithm gives almost the same normalized misfit for all models. Thus, none of the models can be preferred or rejected based on the given data set. Slightly better overall results are obtained for the Kolsky-Futterman model; for one depth zone, the SLS model gave the best result.