Velocity analysis using semblance |

In this appendix, I study the influence of noise on semblance measures. Let us assume that the signal is composed of random independent samples normally distributed with zero mean and variance. In this case, the mathematical expectation for the semblance measure (3) is

Correspondingly, the variance of the noise semblance is

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 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
semblance (7), we deduce that

and

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

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

amean,avari
Mathematical expectation (a) and standard deviation (b) of
random-noise
semblance as a function of the number of
traces
. Solid lines are theoretical curves, circles are
measurements from a numerical experiment.
Figure 11. |
---|

Velocity analysis using semblance |

2013-03-02