^{1, 2}

^{1, 2}

^{3}

^{1}

^{2}

^{3}

We developed a higher resolution method for the estimation of the three travel-time parameters that are used in the 2D zero-offset, Common-Reflection-Surface stack method. The underlying principle in this method is to replace the coherency measure performed using semblance with that of MUSIC (multiple signal classification) pseudospectrum that utilizes the

Many important tasks in seismic processing and imaging require the estimation of travel-time parameters. Such parameters include, among others,

To assess how well a moveout, defined by some trial parameters, approximates a target signal, a number of quantifiers (or

Adopting the notation as in [

Time window used to compute semblance. The two red lines show the travel-time trajectories bounding the window used to select the data.

Semblance can be described in terms of the covariance matrix of the data. Following, for example, [

Even though semblance is a good measure of coherency, it can in many times provide insufficient resolution for the parameter estimation. That is the case, in particular, for interfering events. There is, thus, a motivation to look for alternatives to overcome these difficulties. Attempts have been made to further improve semblance by using only those parts of the data with higher resolving power [

As recognized in sonar and radar applications, methods exploiting the properties of the eigenstructure (namely, eigenvalues and eigenvectors) of the data covariance matrix can lead to far better resolution results than semblance [

This work can be seen as a followup of [

In its original or classical form [

For narrowband signals

For MUSIC to be applicable in our parameter search problem, the different source pulses,

As indicated above, the MUSIC algorithm was originally developed for narrowband and uncorrelated signal applications. If the condition of uncorrelated signals is maintained, an alternative to this situation is to decompose a wideband data into narrowband data components and then treat each narrowband separately [

Seismic signals are highly correlated and require a special modification to be used by the original MUSIC algorithm. The consequence of having correlated sources is that there will be a rank deficiency in the source covariance matrix

In order to handle correlated sources, spatial smoothing over the covariance matrix, can be employed [

Concept of spatial smoothing.

To be able to implement spatial smoothing within seismics, one has to taper the data within a window following the event(s) (cf. Figure

The other advantage of performing the analysis in a given window is to make the steering vectors, required for generating the MUSIC pseudospectrum, to be frequency independent. This allows us to handle wideband seismic data. This process of windowing the event can also be interpreted as steering of the correlation matrix before eigendecomposition and using unity steering vectors for generating the MUSIC pseudospectrum [

Ideally, when the window is “perfectly” matching the event, which will be the case of an optimal choice of the moveout parameters, the signal would be flattened and all traces will nearly have the same moveout. As a consequence, the steering vectors used in (

In practice, the windows are constructed by moveouts, defined by trial parameters. Peaking of the corresponding MUSIC pseudospectra identifies, thus, the “correct” parameters. Following this approach, [

In this section, we compare MUSIC and semblance for travel-time parameter estimation in the situations of classical MUSIC (narrowband uncorrelated signals) and seismic MUSIC (wideband correlated signals). For a simple model of a point diffractor and a dipping reflector with a homogeneous overburden, we analyzed the cases: (a) CMP configuration, which requires the determination of a single parameter,

To illustrate the application of MUSIC for narrowband uncorrelated signals, we considered a point diffractor and a dipping reflector illuminated under a CMP configuration. For a given CMP gather, the data consists of (compare with (

The sources,

Synthetic CMP data used for comparison of MUSIC with semblance. A point diffractor and a dipping reflector (

The output from MUSIC (cf. (

To perform a two-parameter test, we have now simulated a zero-offset (ZO) section for the same previous point diffractor and dipping reflector (cf. Figure

Synthetic ZO section used for comparison of MUSIC with semblance. A point diffractor and a dipping reflector (

Spectrum of parameter

Based on (

Uncorrelated sources: Determination of parameters

Uncorrelated sources: Determination of parameters

To examine the performance of MUSIC compared to semblance in case of wideband correlated signals, we generated synthetic data based on the travel-time (

The results of the parameter search is shown in Figure

Parameter

Correlated sources: Determination of parameters

Correlated sources: Determination of parameters

The first step of the CRS analysis determining the

In order to condition MUSIC to be a normalized quantity, we introduce a scaled version of it, denoted by

A real multioffset GPR data set was used to test out the feasibility of this approach. For an in depth description and discussion of these data, the readers are referred to [

Velocity spectra obtained employing, respectively, semblance (a) and SB-MUSIC (b). The white arrows indicate the apparent single event associated with semblance and the corresponding two events computed from SB-MUSIC.

CMP gather superimposed the hyperbolic moveouts (red curves) for the interfering events based on semblance (a) and SB-MUSIC (b).

In this paper, we discussed the CRS travel-time parameters estimation problem in seismic signal processing. The conventional semblance algorithm was found to generate lower-resolution estimates of the parameters. For the purpose of obtaining higher-resolution parameter estimates, we replaced semblance with MUSIC algorithm. Such procedure allowed us to estimate the parameters within a resolution limit that is significantly better. This work can be seen as a followup of previous applications of MUSIC to single-parameter velocity analysis and slant stacks. Now, MUSIC has been extended to Common-Reflection-Surface (CRS) multiparameter estimation. Applications of the technique to first synthetic examples, consisting of dipping planar reflectors and point diffractors, and comparison to semblance, confirm, at least for these initial situations, the expected far better resolution of MUSIC. To further support this analysis, CMP velocity analysis has been applied to a real multioffset GPR data set. In this situation, better results were obtained upon the introduction of a scaled version of MUSIC, denoted semblance-balanced MUSIC. The new algorithm was seen to outperform semblance in resolving interfering events.

The CRS method uses the so-called generalized hyperbolic (normal) moveout, which is the natural generalization of the NMO, valid for CMP gathers, to CRS supergathers, in which source-receiver pairs are arbitrarily located around the (reference) central point, usually taken as a CMP. In 2D, the generalized hyperbolic moveout depends on three parameters, as opposed to conventional NMO, which depends on a single parameter (NMO velocity).

Mathematically, the generalized hyperbolic moveout,

In the CMP configuration of source-receiver pairs symmetrically located with respect to the central point, namely,

In case the recorded data stems from a diffraction, the condition

CRS parameters for a given source-receiver pair around a central ray at

The authors would like to thank Dr. Hervé Perroud for providing the GPR dataset. E. Asgedom has been funded by a PhD grant from the University of Oslo and the Norwegian Science Foundation. This work has been carried out partly while he was visiting State University of Campinas. M. Tygel acknowledges support of the Brazilian Council of Scientific and Technological Development (CNPq) and the sponsors of the Wave Inversion Technology (WIT) Consortium.