τ-p SEISMIC DATA FOR VISCOELASTIC MEDIA – PART 1: MODELLING1

Viscoelastic modelling reveals that the interaction of compressional-wave velocity C_p, compressional-wave quality factor Q_p, shear-wave velocity C_s, shear-wave quality factor Q_s and Poisson's ratio as a function of time intercept τ and ray parameter p, is complicated; however, distinct, potentially diagnostic behaviours are seen for different combinations of viscoelastic parameters. Synthetic seismograms for three viscoelastic reservoir models show that variations in the Poisson's ratio produce visible differences when compared to the corresponding elastic synthetic seismograms; these differences are attributable to interaction of the elastic parameters with Q_p and Q_s. When the P-wave acoustic impedance contrast is small, viscoelastic effects become more apparent and more useful for interpretation purposes. The corresponding amplitude and net phase spectra reveal significant differences between the elastic and the viscoelastic responses. When P-wave reflectivities are large, they tend to dominate the total response and to mask the Q reflectivity effects. The attenuation effects are manifested as an amplitude decay that increases with both time and ray parameter. The sensitivity of the computed seismic responses for various combinations of viscoelastic parameters suggests the opportunity for diagnostic interpretation of τ-p seismic data. The interpretation of the viscoelastic parameters can permit a better understanding of the rock types and pore fluid distribution existing in the subsurface.     

INVERSION METHODS FOR τ-p MAPS OF NEAR OFFSET DATA—LINEAR INVERSION*

Conventional velocity analysis, based on the ideas of rms velocity and hyperbolic reflection events in the x-t domain, is restricted in validity to near vertical incidence. Thus analysis of near-offset datasets usually requires the muting of wide-angle reflections from shallow interfaces before the rms velocities are determined. The ray-theoretical integral for the delay time τ, which depends on the slowness p and the velocity function, is valid for all angles. The wide-angle reflections can be used to improve the accuracy of the derived velocity function in the near surface region, if the recorded x-t data are mapped into the τ-p domain. By representing the velocity function between reflectors as a series of gradient zones, i.e. regions with a uniform increase in velocity with depth, the recovery of the velocities may be posed as a matrix linear inverse problem for the slopes of the gradient zones. In order to convert the problem to a linear one, the velocity discontinuities at the reflecting interfaces must be fixed in advance. Their positions are based on the behaviour of the τ-p map of the data. Finding a stable velocity model may require several iterations with the reflecting interfaces at different positions. An understanding of the workings of the inversion algorithm allied with an analysis of the causes of instability aids the search for a stable model.     

FREQUENCY WAVENUMBER APPROACH OF THE τ-p TRANSFORM: SOME APPLICATIONS IN SEISMIC DATA PROCESSING*

The τ-p transform is an invertible transformation of seismic shot records expressed as a function of time and offset into the τ (intercept time) and p (ray parameter) domain. The τ-p transform is derived from the solution of the wave equation for a point source in a three-dimensional, vertically non-homogeneous medium and therefore is a true amplitude process for the assumed model. The main advantage of this transformation is to present a point source shot record as a series of plane wave experiments. The asymptotic expansion of this transformation is found to be useful in reflection seismic data processing. The τ-p and frequency-wavenumber (or f-k) processes are closely related. Indeed, the τ-p process embodies the frequency-wavenumber transformation, so the use of this technique suffers the same limitations as the f-k technique. In particular, the wavefield must be sampled with sufficient spatial density to avoid wavenumber aliasing. The computation of this transform and its inverse transform consists of a two-dimensional Fast Fourier Transform followed by an interpolation, then by an inverse-time Fast Fourier Transform. This technique is extended from a vertically inhomogeneous three-dimensional medium to a vertically and laterally inhomogeneous three-dimensional medium. The τ-p transform may create artifacts (truncation and aliasing effects) which can be reduced by a finer spatial density of geophone groups by a balancing of the seismic data and by a tapering of the extremities of the seismic data. The τ-p domain is used as a temporary domain where the attack of coherent noise is well addressed; this technique can be viewed as ‘time-variant f-k filtering’. In addition, the process of deconvolution and multiple suppression in the τ-p domain is at least as well addressed as in the time-offset domain.     

BAYESIAN ESTIMATION IN SEISMIC INVERSION. PART I: PRINCIPLES1

This paper gives a review of Bayesian parameter estimation. The Bayesian approach is fundamental and applicable to all kinds of inverse problems. Its basic formulation is probabilistic. Information from data is combined with a priori information on model parameters. The result is called the a posteriori probability density function and it is the solution to the inverse problem. In practice an estimate of the parameters is obtained by taking its maximum. Well-known estimation procedures like least-squares inversion or l₁ norm inversion result, depending on the type of noise and a priori information given. Due to the a priori information the maximum will be unique and the estimation procedures will be stable except (in theory) for the most pathological problems which are very unlikely to occur in practice. The approach of Tarantola and Valette can be derived within classical probability theory. The Bayesian approach allows a full resolution and uncertainty analysis which is discussed in Part II of the paper.     

BAYESIAN ESTIMATION IN SEISMIC INVERSION. PART II: UNCERTAINTY ANALYSIS1

A parameter estimation or inversion procedure is incomplete without an analysis of uncertainties in the results. In the fundamental approach of Bayesian parameter estimation, discussed in Part I of this paper, the a posteriori probability density function (pdf) is the solution to the inverse problem. It is the product of the a priori pdf, containing a priori information on the parameters, and the likelihood function, which represents the information from the data. The maximum of the a posteriori pdf is usually taken as a point estimate of the parameters. The shape of this pdf, however, gives the full picture of uncertainty in the parameters. Uncertainty analysis is strictly a problem of information reduction. This can be achieved in several stages. Standard deviations can be computed as overall uncertainty measures of the parameters, when the shape of the a posteriori pdf is not too far from Gaussian. Covariance and related matrices give more detailed information. An eigenvalue or principle component analysis allows the inspection of essential linear combinations of the parameters. The relative contributions of a priori information and data to the solution can be elegantly studied. Results in this paper are especially worked out for the non-linear Gaussian case. Comparisons with other approaches are given. The procedures are illustrated with a simple two-parameter inverse problem.     

Multiple suppression by 2D filtering in the parabolic τ–p domain: a wave-equation-based method1

The filter for wave-equation-based water-layer multiple suppression, developed by the authors in the x-t, the linear τ-p, and the f-k domains, is extended to the parabolic τ-2 domain. The multiple reject areas are determined automatically by comparing the energy on traces of the multiple model (which are generated by a wave-extrapolation method from the original data) and the original input data (multiples + primaries) in τ-p space. The advantage of applying the data-adaptive 2D demultiple filter in the parabolic τ-p domain is that the waves are well separated in this domain. The numerical examples demonstrate the effectiveness of such a dereverberation procedure. Filtering of multiples in the parabolic τ-p domain works on both the far-offset and the near-offset traces, while the filtering of multiples in the f-k domain is effective only for the far-offset traces. Tests on a synthetic common-shot-point (CSP) gather show that the demultiple filter is relatively immune to slight errors in the water velocity and water depth which cause arrival time errors of the multiples in the multiple model traces of less than the time dimension (about one quarter of the wavelet length) of the energy summation window of the filter. The multiples in the predicted multiple model traces do not have to be exact replicas of the multiples in the input data, in both a wavelet-shape and traveltime sense. The demultiple filter also works reasonably well for input data contaminated by up to 25% of random noise. A shallow water CSP seismic gather, acquired on the North West Shelf of Australia, demonstrates the effectiveness of the technique on real data.     

COMMON UNITGRAPHS FOR SETS OF RUNOFF EVENTS. PART 1: UNITGRAPH IDENTIFICATION FROM STREAMFLOW DATA

It is demonstrated that a unitgraph can be obtained without using rainfall data, provided data from at least two runoff events are available. A numerical method has been devised which calculates this common unitgraph for a set of surface runoff events and at the same time determines the input (rainfall excess) for each event. The method has been successfully tested on catchments ranging in size from 0.4 to 600 km²; it requires streamflow observations taken at intervals which retain all significant frequencies in the runoff hydrographs. The method also requires application of a baseflow separation procedure which is consistent for all events. The new approach has the potential to lead to more objective studies of the effects of catchment changes on the unitgraph and provides scope for comparisons of the common unitgraphs with geomorphological instantaneous unitgraphs.     

COMPARISON OF “AQUAFLEX” WITH CONVENTIONAL EXPLOSIVE FOR SEISMIC REFRACTION MEASUREMENTS AT SEA*

A seismic refraction line fired using conventional explosive charges was repeated using a seismic cord, “Aquaflex”. It was found that travel times and frequencies were the same for both kinds of source and that amplitudes from “Aquaflex” were comparable with those produced using conventional explosives more than ten times the weight. This advantage carries with it gains in cost and efficiency and reduces damage to fishing.     

Magmatismes basiques dits “orogeniques” et “anorogeniques” et teneurs entro2: Les associations “Isotitanees” et “anisotitanfes”

The patterns of variation of TiO₂ conent during magmatic evolution are different in the so called “orogenic” and “anorogenie” basic associations; these last terms, which are the cause of much misunderstanding, can be replaced by the terms “isotitaniferous” and “anisotitaniferous”.     

Practical and robust experimental determination of c13 and Thomsen parameter δ

We analysed the complications in laboratory velocity anisotropy measurement on shales. There exist significant uncertainties in the laboratory determination of c₁₃ and Thomsen parameter δ. These uncertainties are primarily related to the velocity measurement in the oblique direction. For reliable estimation of c₁₃ and δ, it is important that genuine phase velocity or group velocity be measured with minimum uncertainty. The uncertainties can be greatly reduced if redundant oblique velocities are measured. For industrial applications, it is impractical to make multiple oblique velocity measurements on multiple core plugs. We demonstrated that it is applicable to make multiple genuine oblique group velocity measurements on a single horizontal core plug. The measurement results show that shales can be classified as a typical transversely isotropic medium. There is a coupling relation between c₄₄ and c₁₃ in determining the directional dependence of the seismic velocities. The quasi‐P‐wave or quasi‐S‐wave velocities can be approximated by three elastic parameters.     

Physical constraints on c13 and δ for transversely isotropic hydrocarbon source rocks

Based on the theory of anisotropic elasticity and observation of static mechanic measurement of transversely isotropic hydrocarbon source rocks or rock‐like materials, we reasoned that one of the three principal Poisson's ratios of transversely isotropic hydrocarbon source rocks should always be greater than the other two and they should be generally positive. From these relations, we derived tight physical constraints on c₁₃, Thomsen parameter δ, and anellipticity parameter η. Some of the published data from laboratory velocity anisotropy measurement are lying outside of the constraints. We analysed that they are primarily caused by substantial uncertainty associated with the oblique velocity measurement. These physical constraints will be useful for our understanding of Thomsen parameter δ, data quality checking, and predicting δ from measurements perpendicular and parallel to the symmetrical axis of transversely isotropic medium. The physical constraints should also have potential application in anisotropic seismic data processing.     

The ρ-family of algorithms for time-step integration with improved numerical dissipation

The dynamic analysis of complex non-linear structural systems by the finite element approach requires the use of time-step algorithms for solving the equations of motion in the time domain. Both an implicit and an explicit version of such a time-step algorithm, called the ρ-method, the parameter ρ being used for controlling numerical damping in the higher modes, are presented in this paper. For the implicit family of algorithms unconditional stability, consistency, convergence, accuracy and overshoot properties are first discussed and proved. On the basis of the algorithmic damping ratio (dissipation) and period elongation (dispersion) the ρ-method is then compared with the well-known implicit algorithms of Hilber, Newmark, Wilson, Park and Houbolt. An explicit version of the algorithm is also derived and briefly discussed. This shows numerical properties similar to the central difference method. Both versions of the algorithm have been implemented in a general purpose computer program which has been often used for both numerical tests and practical applications.     

Influence of Surfactants on Degradation of 1‐(2‐Chlorobenzoyl)‐3‐(4‐chlorophenyl) Urea by Nanoscale Zerovalent Iron

Nanoscale zerovalent iron (NZVI) has been proved to be effective in the degradation of environmental pollutants and exhibits advantages in the removal of 1‐(2‐chlorobenzoyl)‐3‐(4‐chlorophenyl) urea (CCU), an analog of diflubenzuron. This present study focused on the influence of surfactants in the degradation procedure with NZVI in order to provide a simple and rapid removal method for CCU. Triton X‐100, Tween 20, Tween 80, sodium dodecyl sulfonate (SDS), and cetyltrimethylammonium bromide (CTAB) were investigated under anaerobic conditions. The experimental results demonstrated that the degradation rate increased sharply with the presence of SD during the first 15 min, up to 99.97% with addition of 0.01 g L^?1 SDS, whereas the presence of Triton X‐100, Tween 80, and Tween 20 resulted in a slight enhancement of the degradation of CCU. The enhancement strength of them was in the order Tween 20, Triton X‐100, and Tween 80. However, addition of the cationic surfactant CTAB resulted in a significant inhibitive effect. In contrast, the mixed surfactants did not result in the expected performance, and the performance was lower than that using some certain single surfactant among the mixed surfactants.     

Factors Affecting UV/H2O2 Oxidation of 17α‐Ethynyestradiol in Water

In this study, bench‐scale experiments were conducted to examine the UV/H₂O₂ oxidation of 17α‐ethynyestradiol (EE2) in water in a batch operation mode. The EE2 degradation exhibited pseudo‐first‐order kinetics, and the removal was ascribed to the production of hydroxyl radicals (^?OH) by the UV/H₂O₂ system. Typically, the EE2 oxidation rate increased with increasing UV intensity and H₂O₂ dose, and with deceasing initial EE2 levels and solution pH. At EE2₀ = 650 µg/L, UV intensity = 154 µW/cm², H₂O₂ = 5 mg/L, and neutral pH, the UV/H₂O₂ treatment was able to remove 90% of the EE2 content within 30 min. Four anions commonly present in water were found to inhibit EE2 degradation to varying degrees: > > Cl^? > . Our results demonstrate that the described UV/H₂O₂ process is an effective method to control EE2 pollution in water.