Apparatus for mode extraction using multiple frequencies

1. An apparatus that extracts slowness dispersion characteristics of one or more sonic wave forms in broadband acoustic waves received by an array of two or more sensors positioned with respect to a reservoir, the apparatus comprising:

a circuit that digitizes the sonic wave forms to form discrete time wave forms and converts the discrete time wave forms into frequency domain wave forms data;

a circuit that divides a processing band of the frequency domain wave forms data into frequency sub-bands data;

one or more processors that, for each sub-band:

approximate a family of candidate dispersion curves for multiple modes;

parameterize each of the family of approximated candidate dispersion curves by phase and group slowness;

form a frequency dependent over-complete dictionary of basis elements, each corresponding to a pair of phase and group slownesses, and spanning a range of values thereof;

form multiple measurement vectors from the frequency domain wave forms data;

implement a sparse Bayesian learning (SBL) algorithm on the multiple measurement vectors with a block sparse signal model by:

estimation of a mode spectrum amplitude with a Bayesian approach;

estimation of a mode spectrum variance, noise variance and other model parameters with a maximum likelihood estimation;

assessment of presence of spectrum correlation patterns of multiple modes in the frequency domain wave forms data based in part on an estimation of covariance therein;

prune mode candidates based at least in part on the assessment and by comparison of the estimated mode parameters with a threshold;

iteration between the Bayesian approach and the maximum likelihood estimation until convergence according to a criterion; and

output the phase and group slowness and the mode spectrum amplitude for a given sub-band; and that

generate a final dispersion curve over the processing band; and

characterize the reservoir based at least in part on the final dispersion curve.

2. The apparatus according to claim 1 , wherein; the Bayesian minimum mean square estimation and/or the Bayesian maximum a posteriori estimation is implemented during the broadband SBL algorithm.

3. The apparatus according to claim 1 , wherein; the maximum marginal likelihood estimation or the Type-II maximum likelihood estimation is implemented during the broadband SBL algorithm.

4. The apparatus according to claim 1 , wherein; the broadband SBL algorithm uses a fixed-point (FP) algorithm.

5. The apparatus according to claim 1 , wherein; the broadband SBL algorithm uses an Expectation-Maximization (EM) algorithm.

6. The apparatus according to claim 1 , wherein; the assessment of presence of spectrum correlation in the SBL algorithm is achieved by a means of joint Gaussian distributions and/or conditional stochastic distributions.

7. The apparatus according to claim 6 , wherein; the conditional stochastic distributions include a conditional uniform distribution and a truncated Gaussian distribution.

8. The apparatus according to claim 1 , wherein: the broadband SBL algorithm uses a global workflow and a peak estimating function applied over each frequency band.

9. The apparatus according to claim 1 , wherein: the broadband SBL algorithm uses a local workflow and a peak estimating function applied over a whole frequency range.

10. The apparatus according to claim 1 , wherein: the broadband SBL algorithm uses a correlation matrix structure for the parameter estimation.

11. The apparatus according to claim 10 , wherein: the correlation matrix structure includes a Hermitian, Toeplitz, persymmetric, band-limited structures.

12. The apparatus according to claim 1 , wherein: the prune is implemented at least in part by thresholdation that outs the modes with estimated variances and/or amplitudes of less than a certain value.

13. A method for extracting slowness dispersion characteristics of one or more sonic wave forms in broadband acoustic waves received by an array of two or more sensors positioned with respect to a reservoir, the method comprising:

digitizing the sonic wave forms to form discrete time wave forms and converting the discrete time wave forms into frequency domain wave forms;

dividing a processing band of the frequency domain wave forms into frequency sub-bands; and

for each sub-band:

approximating a family of candidate dispersion curves for multiple modes;

parameterizing each of the family of approximated candidate dispersion curves by phase and group slowness;

forming a frequency dependent over-complete dictionary of basis elements, each corresponding to a pair of phase and group slownesses, and spanning a range of values thereof;

forming multiple measurement vectors from the frequency domain data;

via one or more processors, implementing a sparse Bayesian learning (SBL) algorithm on the multiple measurement vectors with a block sparse signal model by:

estimating a mode spectrum amplitude with a Bayesian approach;

estimating a mode spectrum variance, noise variance and other model parameters with a maximum likelihood estimation;

assessing presence of spectrum correlation patterns of multiple modes;

pruning mode candidates by comparing the estimated mode parameters with a threshold;

iterating between the Bayesian approach and the maximum likelihood estimation until convergence according to a criterion; and

outputting the phase and group slowness and the mode spectrum amplitude for a given sub-band; and

generating a final dispersion curve over the processing band; and

characterizing the reservoir based at least in part on the final dispersion curve.