Local seismic attributes |

Let represent seismic trace as a function of time . The
corresponding complex trace is defined as

By definition, instantaneous frequency is the time derivative of the instantaneous phase (Taner et al., 1979)

Different numerical realizations of equation 3 produce slightly different algorithms (Barnes, 1992).

Note that the definition of instantaneous frequency calls for division
of two signals. In a linear algebra notation,

where stands for the identity operator. Stabilization by does not, however, prevent instantaneous frequency from being a noisy and unstable attribute. The main reason for that is the extreme locality of the instantaneous frequency measurement, governed only by the phase shift between the signal and its Hilbert transform.

Figure shows three test signals for comparing frequency attributes. The first signal is a synthetic chirp function with linearly varying frequency. Instantaneous frequency shown in Figure 1 correctly estimates the modeled frequency trend. The second signal is a piece of a synthetic seismic trace obtained by convolving a 40-Hz Ricker wavelet with synthetic reflectivity. The instantaneous frequency (Figure 1b) shows many variations and appears to contain detailed information. However, this information is useless for characterizing the dominant frequency content of the data, which remains unchanged due to stationarity of the seismic wavelet. The last test example (Figure c) is a real trace extracted from a seismic image. The instantaneous frequency (Figure 1c) appears noisy and even contains physically unreasonable negative values. Similar behavior was described by White (1991).

inst
Instantaneous frequency of test
signals from Figure .
Figure 1. |
---|

locl
Local frequency of test signals
from Figure .
Figure 2. |
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Local seismic attributes |

2013-07-26