Seismic attenuation values obtained from instantaneous-frequency matching and spectral ratios

In this paper, attenuation values are obtained from seismic data using instantaneous-frequency matching and spectral ratios. to obtain differential t * values using instantaneous-frequency matching, a near offset reference pulse is attenuated until the resulting instantaneous frequency matches the observed value at the receiver. Prior to matching, filtering can be applied to each trace in order to reduce the effects of noise on the calculated instantaneous frequencies. In the second method, the spectral ratio between a receiver pulse and a reference pulse is used to obtain differential t * values. to obtain an unbiased estimate, a variable spectral bandwidth is used depending on the noise level of the data. the two methods are tested using synthetic traces and then applied to crustal refraction data from the 1986 PASSCAL Ouachita experiment. Results show that the differential t * values obtained using filtered, instantaneous-frequency matching are consistent with and have less scatter than those obtained from spectral ratios with a variable bandwidth.     

Seismic reflection and transmission coefficients of a self-similar interface

The derivation of seismic reflection and transmission coefficients is generally based on the assumption that the medium parameters behave as step functions of depth, at least in a finite region around the interface. However, outliers observed in well logs generally behave quite differently from step functions. In this paper we represent an interface by a self-similar singularity, embedded between two homogeneous half-spaces, and we derive its frequency-dependent normal-incidence reflection and transmission coefficients. For ω  → 0 the expressions for the coefficients reduce to those for a discrete boundary between two homogeneous half-spaces; for ω → ∞ they become frequency-independent. These asymptotic expressions have a relatively simple form and depend on the singularity exponent α .
  The exact as well as the asymptotic expressions are used to evaluate the time-domain reflection and transmission responses of a self-similar interface. Finally, we use a numerical method to model the response of a smoothed version of a self-similar interface (note that the velocity of a smoothed singularity remains finite). It turns out that smoothing has hardly any effect on the response, provided that the smoothing does not affect the scales corresponding to the seismic frequency range.     

Seismic reflection traveltimes in two-dimensional statistically anisotropic random media

Velocity estimation remains one of the main problems when imaging the subsurface with seismic reflection data. Traveltime inversion enables us to obtain large-scale structures of the velocity field and the position of seismic reflectors. However, as the media currently under study are becoming more and more complex, we need to know the finer-scale structures. The problem is that below a certain range of velocity heterogeneities, deterministic methods become difficult to use, so we turn to a probabilistic approach. With this in view, we characterize the velocity field as a random field defined by its first and second statistical moments. Usually, a seismic random medium is defined as a homogeneous velocity background perturbed by a small random field that is assumed to be stationary. Thus, we make a link between such a random velocity medium (together with a simple reflector) and seismic reflection traveltimes. Assuming that the traveltimes are ergodic, we use 2-D seismic reflection geometry to study the decrease in the statistical traveltime fluctuations as a function of the offset (the source–receiver distance). Our formulae are based on the Rytov approximation and the parabolic approximation for acoustic waves. The validity and the limits are established for both of these approximations in statistically anisotropic random media. Finally, theoretical inversion procedures are developed for the horizontal correlation structure of the velocity heterogeneities for the simplest case of a horizontal reflector. Synthetic seismograms are then computed (on particular realizations of random media) by simulating scalar wave propagation via finite difference algorithms. There is good agreement between the theoretical and experimental results.     

Two-dimensional PP/PS-stereotomography: P- and S-waves velocities estimation from OBC data

We present the extension of stereotomography to P - and S -wave velocity estimation from PP - and PS -reflected/diffracted waves. In this new context, we greatly benefit from the use of locally coherent events by stereotomography. In particular, when applied to S -wave velocity estimation from PS -data, no pairing of PP - and PS -events is a priori required. In our procedure the P -wave velocity model is obtained first using stereotomography on PP -arrivals. Then the S -wave velocity model is obtained using PS -stereotomography on PS -arrivals fixing the P -wave velocity model. We present an application to an 'ideal' synthetic data set demonstrating the relevance of the approach, which allows us to recover depth consistent P - and S -waves velocity models even if no pairing of PP - and PS -events is introduced. Finally, results to a real data set from the Gulf of Mexico are presented demonstrating the potential of the method in a noisy data context.