# Dictionary:Instantaneous frequency

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The instantaneous frequency is a seismic attribute, which is defined to be the time rate of change of the instantaneous phase. Computationally it is preferable to compute the time rate of change as the derivative of the arctangent function to avoid ${\displaystyle 2\pi }$ discontinuities. The computed output is given in units of cycles per second. Instantaneous frequencies are influenced by the bed thickness and we wish to observe them without too much interference. This is accomplished by using several adjacent traces to form a consistent output. It has been shown than instantaneous frequency, computed as the time derivative of instantaneous phase relates the the centroid of the power spectrum of the seismic wavelet. [1]

${\displaystyle Freq(x,t)={\frac {d[Ph(x,t)]}{d(t)}}={\frac {f(x,t)\cdot \displaystyle {\frac {dg}{dt}}-g(x,t)\cdot \displaystyle {\frac {df}{dt}}}{f^{2}(x,t)+g^{2}(x,t)}}}$

Where ${\displaystyle Ph(x,t)}$ is the instantaneous phase calculation and ${\displaystyle f(t)}$ is the real part of the seismic data and ${\displaystyle g(t)}$ is the imaginary part of the complex trace i.e the Hilbert transform of ${\displaystyle f(t)}$. Instantaneous frequencies relate the wave propagation and depositional environment, hence they are physical attributes and can be used as effective discriminators:

• Corresponds to the average frequency (centroid) of the power spectrum of the seismic wavelet.
• Seismic character correlator in lateral direction.
• Indicates the edges of low impedance thin beds
• Hydrocarbon indicator by low frequency anomaly. This effect is sometimes accentuated by unconsolidated sands due to oil content of the pores.
• Fracture zone indicator, they may appear as lower frequency zones.
• Chaotic reflection zone indicator, due to excessive scatter.
• Bed thickness indicator. Higher frequencies indicate sharp interfaces or thin shale bedding, lower frequencies indicate sand rich bedding.
• Sand/Shale ratio indicator in a clastic environment.

## References

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