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Next: Appendix B: Automatic velocity Up: Fomel: Velocity analysis using Previous: Acknowledgments

Apendix A: Statistical analysis of semblance measures

In this appendix, I study the influence of noise on semblance measures. Let us assume that the signal $ \mathbf{a}=a_1,a_2,\ldots,a_N$ is composed of random independent samples normally distributed with zero mean and $ \sigma^2$ variance. In this case, the mathematical expectation for the semblance measure (3) is

$\displaystyle E\left[\beta^2(\mathbf{a})\right] = \frac{\displaystyle E\left[\l...
...\right]} {\displaystyle N\,\sum_{i=1}^{N} E\left[a_i^2\right]} = \frac{1}{N}\;.$ (9)

Correspondingly, the variance of the noise semblance is
$\displaystyle V\left[\beta^2(\mathbf{a})\right]$ $\displaystyle =$ $\displaystyle \frac{\displaystyle E\left[\left(\sum_{i=1}^N a_i\right)^4\right]...
...sum_{i=1}^{N} \sum_{j \ne i} E\left[a_i^2\,a_j^2\right]\right)} -
\frac{1}{N^2}$  
  $\displaystyle =$ $\displaystyle \frac{3\,N\,\sigma^4 + 3\,(N^2-N)\,\sigma^4}
{N^2\,\left[3\,N\,\s...
...4 + (N^2-N)\,\sigma^4\right]} -
\frac{1}{N^2} = \frac{2\,(N-1)}{N^2\,(N+2)}\;.$ (10)

Equations (A-1) and (A-2) show that both the mathematical expectation and the standard deviation (the square root of variance) of the random noise semblance decrease at the rate of $ 1/N$ with the increase in the number of traces. To derive these equations, I make an assumption that the terms in the numerator and denominator are statistically independent. Rather than proving this assumption mathematically, I test it by numerical experiments with multiple random number realizations. Figure A-1 compares the theoretical prediction with experimental measurements from 10,000 random realizations.

Applying similar analysis to the $ AB$ semblance (7), we deduce that

$\displaystyle E\left[\alpha^2(\mathbf{a})\right]$ $\displaystyle =$ $\displaystyle \frac{\displaystyle E\left[2\,\sum_{i=1}^{N} a_i\,\sum_{i=1}^N \p...
...]\,
\left[\left(\sum_{i=1}^N \phi_i\right)^2 - N\,\sum_{i=1}^N \phi_i^2\right]}$  
  $\displaystyle =$ $\displaystyle \frac{\displaystyle 2\,E\left[a_i^2\right]\,\left[\left(\sum_{i=1...
...um_{i=1}^N \phi_i\right)^2 - N\,\sum_{i=1}^N \phi_i^2\right]}
= \frac{2}{N}\;.$ (11)

and
$\displaystyle V\left[\alpha^2(\mathbf{a})\right]$ $\displaystyle =$ $\displaystyle \frac{\displaystyle E\left[\left\{2\,\sum_{i=1}^{N} a_i\,\sum_{i=...
...um_{i=1}^N \phi_i\right)^2 - N\,\sum_{i=1}^N \phi_i^2\right]^2} - \frac{4}{N^2}$  
  $\displaystyle =$ $\displaystyle \frac{\displaystyle 2\,\left[2\,\left(\sum_{i=1}^N \phi_i\right)^...
...um_{i=1}^N \phi_i\right)^2 - N\,\sum_{i=1}^N \phi_i^2\right]^2} - \frac{4}{N^2}$  
  $\displaystyle =$ $\displaystyle \frac{4\,(N-2)}{N^2\,(N+2)}\;.$ (12)

One can see that, in the case of the $ AB$ semblance, the mathematical expectation and the standard deviation of the random noise semblance decrease at the rate of $ 2/N$ , twice higher than that for the conventional semblance. Figure A-2 compares the theoretical prediction with experimental measurements.

mean vari
mean,vari
Figure 10.
Mathematical expectation (a) and standard deviation (b) of random-noise semblance as functions of the number of traces $ N$ . Solid lines are theoretical curves, circles are measurements from a numerical experiment.
[pdf] [pdf] [png] [png] [scons]

amean avari
amean,avari
Figure 11.
Mathematical expectation (a) and standard deviation (b) of random-noise $ AB$ semblance as a function of the number of traces $ N$ . Solid lines are theoretical curves, circles are measurements from a numerical experiment.
[pdf] [pdf] [png] [png] [scons]


next up previous [pdf]

Next: Appendix B: Automatic velocity Up: Fomel: Velocity analysis using Previous: Acknowledgments

2013-03-02