# Dictionary:Instantaneous bandwidth

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The instantaneous bandwidth is a seismic attribute. The are a two methods to calculate instantaneous bandwidth one from Barnes (1992) and the other from O'Doherty (1992).

Barnes' equation is seen here:

${\displaystyle \sigma (t)=\left|{\frac {\displaystyle {\frac {dE(t)}{dt}}}{\displaystyle 2\pi E(t)}}\right|}$

Where ${\displaystyle E(t)}$ is the trace envelope. This equation measures the absolute value of the rate of change of the trace envelope amplitude.

O'Doherty's equation involves the centroid of the power spectrum of a wavelet ${\displaystyle \omega _{c}}$ , the variance ${\displaystyle \omega _{v}}$, and the RMS frequency ${\displaystyle \omega _{RMS}}$.

The frequency corresponding to the centroid of the power spectrum of a wavelet can be calculated here:

${\displaystyle \omega _{c}={\frac {\displaystyle \int _{0}^{\infty }\omega P(\omega )d\omega }{\displaystyle \int _{0}^{\infty }P(\omega )d\omega }}}$

The variance with respect to the centroid frequency is given by:

${\displaystyle \omega _{v}^{2}={\frac {\displaystyle \int _{0}^{\infty }(\omega -\omega _{c})^{2}P(\omega )d\omega }{\displaystyle \int _{0}^{\infty }P(\omega )d\omega }}}$

The RMS frequency, also known as the second moment of the power spectrum, can be calculated by:

${\displaystyle \omega _{RMS}={\frac {\displaystyle \int _{0}^{\infty }\omega ^{2}P(\omega )d\omega }{\displaystyle \int _{0}^{\infty }P(\omega )d\omega }}}$

By expanding the variance equation we can show that:

${\displaystyle \omega _{v}^{2}=\omega _{RMS}^{2}-\omega _{c}^{2}}$

Now we can examine these statistical measurements of the power spectrum in the form of useful attributes. These computations represent the statistics of the seismic wavelet computed over some time window. Therefore, they are more closely associated with the time smoothed instantaneous attributes. The are computed and displayed continuously for all the data samples. By definition ${\displaystyle \omega _{c}}$, the centroid frequency is the mean frequency where an equal amount of energy exists on either side of this frequency. The variance with respect to the mean frequency indicates the width of the power spectral density distribution over a band of frequencies, therefore we can use it as an indication of the spectral bandwidth. [1]

Instantaneous bandwidth is a statistical measure of the seismic wavelet, but it relates to various physical conditions:

• Represents seismic data bandwidth sample by sample. It is one of the high-resolution character correlators.
• Shows overall effects of absorption and seismic character changes.

## References

1. Taner, Turhan (1992), Attributes Revisited, Rock Solid Images Houston, Texas (published 2000)