A Processing Technique to Determine Slowness of Multiple Acoustic Wave Modes Propagating in Attenuated Media, 14-R9700

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Principal Investigators
Jorge O. Parra
Chris L. Hackert
Dawn Domaschk
P.-C. Xu

Inclusive Dates:  04/01/07 – Current

Background - The predictive method developed by X.M. Tang can effectively process borehole array waveforms and obtain slowness from mixed multiple wave modes. In this method, the waveform at a receiver is modeled by a combination of wave data from other receivers of the array in terms of the time delays due to receiver offsets and modal slownesses. The inversion of the unknown slownesses is then carried out through the minimization of an error function that covers the entire array data set and represents the difference between the predicted and measured wave data. The predictive method in general works well when the attenuation is absent. However, under certain conditions, such as mixed weak and strong wave modes, the error function tends to miss or dislocate some slownesses. In addition, the commonly used Levenberger-Marquardt algorithm for minimization requires the derivatives of the error function with respect to the slownesses, which must be obtained numerically. This step not only adds to the processing time but also increases chances of numerical errors. Most importantly, the predictive method of the original version is not valid for attenuated media. Therefore, the objectives of this project are: revise the definition of the error function to more directly and accurately target the unknown slownesses; extend the method to include attenuation; and replace the Levenberger-Marquardt routine by another minimization algorithm that does not need derivatives of the error function.

Approach - In the original predictive method, the error function involves a summation over both time within the window and receivers across the array. To extend the predictive method to the viscoelastic case, a Fourier transform of the error function is used. In the frequency domain, factors in each term are multiplied to recover the amplitude change due to attenuation. The new error function contains the unknown slowness plus quality factor for all wave modes and involves a summation over both frequency in the spectrum and receivers across the array. The quality factors of the P- and S- waves can be estimated using SwRI's recently developed processing algorithm.The quality factor of a guided mode can be expressed in terms of the quality factors of the P- and S- waves, and partial derivatives of the error functions with respect to all slownesses. Finally, the yet unknown quality factors of guided wave modes can be determined using such a relation along with the inversion of the unknown slownesses in an iterative procedure.

In Tang's method, the predictive function is constructed from waveforms recorded at receivers both above and below the target receiver. This is termed two-way predicting. In one-way predicting, only receivers above or below are used. After examining properties of the predicted function for many sets of model parameters, the error function surface produced by one-way predicting has a nicer shape for minima picking, and the minima locations (leaky P-wave and flexural wave) are more accurate.

It is quite common in a monopole seismogram that the P head wave or Stoneley wave are much weaker than the pseudo-Rayleigh wave. Similarly, in a dipole seismogram, the S head wave (for fast formations) can be weak, too. Both cases involve a weak event of interest along with stronger events. The resultant error function often does not have a well developed and distinct minimum. As a result, it is difficult for any minimization algorithm to obtain that slowness successfully. To improve the situation, a logarithmic normalization algorithm is proposed to allow all minima to be well developed, outstanding, and correctly located

The Simplex algorithm does not need derivates of the error function and hence is a good candidate to replace the Levenberger-Marquardt algorithm. However, the Simplex algorithm needs to cooperate with an automatic scanning scheme to pinpoint the global minima.

Accomplishments - Mathematical developments for the extension of the predictive method to the viscoelastic case are complete. In this aspect, most subroutines have been or are being completed. One-way predicting was formulated, implemented and tested, as was the logarithmic normalization. A multi-dimensional Simplex algorithm was also implemented. An automatic scanning scheme to support the Simplex algorithm is being developed. At this stage, some preliminary results have been obtained for slowness processing. Model parameters are given in Table 1. The reference values of slownesses for these models from their dispersion curves are listed in Table 2. Finally, the inversion results with accuracy rate are given in Table 3. It can be seen that the accuracy is good.

Table 1. Model parameters

  D (m) Vf (km/s) ρf (g/cm3) ρ (g/cm3) VP (km/s) VS (km/s)
1 0.25 1.5 1.0 2.55 3.50 1.95
2 0.25 1.5 1.0 1.2 2.70 1.12

 

Table 2. Reference values of slowness for major wave modes (ms/m)

  P wave Pseudo-Rayleigh
wave @10 kHz
Stoneley wave
@10 kHz
S wave Flexural wave
@3 kHz
1 0.286 0.556 0.709 0.513 0.524
2 0.370 N/A 1.103 0.893 1.013

 

Table 3. Slowness (ms/m) obtained by the current processing algorithm using good input parameters

  P head wave Pseudo-Rayleigh
wave
Stoneley
wave
S head
wave
Flexural
wave
1 0.300 0.550 0.683 0.487* 0.487*
4.9 % 1.9 % 3.7 % 5.1 % 7.1 %
2 0.377 N/A 1.043 N/A 1.029
1.9 % N/A 5.4 % N/A 1.6 %
* Because the S head wave and flexural wave slownesses are very close for Model 1, the processing algorithm finds only one value.

 


Figure 1. The original error function surface for model 2 when the source is a monopole.
 


Figure 2. The logarithmically normalized error function surface for model 2 when the source is a monopole.

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