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Nonstationary Signals and Spectral Properties

Stationary signals are any idealised signals that have time-independent properties, such as time period, frequency, and spectral content. Seismic signals, however, are nonstationary signals that violate the stationary rule described above. When seismic waves propagate through the anelastic media in the Earth's subsurface, seismic signals have variable properties that vary with the propagation path and travel time.

A stationary signal can be represented mathematically by a stack of sinusoids, using the Fourier transform. In contrast, a nonstationary signal cannot be properly represented by the Fourier transform because the amplitude and frequency of a sinusoidal representation can change dynamically as a function of travel time. To investigate the local properties of a nonstationary signal, an analytic signal-based analysis method can be used instead.

1.1 Stationary Signals

For seismic signals, an idealised stationary model is a stationary convolution process. The physical meaning of the convolution process is 'superposition', which is a fundamental principle in the analysis of seismic signals.

Consider an earth model with a layered structure (Figure 1.1). Each layer has a different acoustic impedance, which is the product of velocity and density. The contrast of this physical property between two layers causes seismic reflections at the interface.

Figure 1.1 The principle of 'superposition'. Each wavelet is scaled by the reflectivity, which is the contrast in the acoustic impedance between two layers. A seismic trace is formed by summing all scaled wavelets, which are reflected from various interfaces.

The contrast of acoustic impedance is called reflectivity, or reflection coefficient. The seismic reflectivity includes primary and multiple , where is the travel time. The reflectivity series in time consists of both types, . Each reflectivity serves as a scaling factor to scale a particular wavelet . A recorded seismic trace is formed by summing all scaled wavelets reflected from different interfaces.

This physical process is the 'superposition' and can be written as

where is the vector of the reflectivity series, is the vector of the seismic wavelet, and is the vector of the seismic trace. If the length of the reflectivity series is , and the length of the seismic wavelet is , the length of the resulting seismic trace is .

In the superposition process of Eq. (1.1), the wavelet is time shifted and scaled by the reflectivity. Combining all the time-shifted wavelets to form a wavelet matrix, , the superposition process can be expressed in a matrix-vector form, , and written explicitly as

in which the size of the wavelet matrix is .

In the matrix-vector form of Eq. (1.2), the wavelet matrix is a Toeplitz matrix, because each diagonal of is a constant, . Then, if we look at the wavelet matrix row by row, we can see that each row of the matrix consists of the discretised wavelet samples in time-reversed order:

Therefore, the matrix-vector form of superposition, , is the discretised form of 'convolution'. The corresponding continuous form for the convolution process can be expressed as

where is the seismic wavelet, is the subsurface reflectivity series, is the convolution operator, and is the seismic trace. In this convolution process, the convolution kernel is the stationary wavelet , which has a constant time period, frequency, and spectral content, with respect to time variation.

A wavelet is a 'small wave', for which the time period is relatively short compared to the time duration of the reflectivity series and the resulting seismic trace. With the stationary wavelet , the stationary convolution process of Eq. (1.4) can be depicted as in Figure 1.2.

The stationary wavelet, that is used in Figure 1.2 for the demonstration, is the Ricker wavelet defined as (Ricker, 1953)

where is the dominant frequency (in ), and is the central position of the wavelet. The Ricker wavelet is symmetrical with respect to time and has a zero mean, . Therefore, it is often known as Mexican hat wavelet in the Americas for its sombrero shape.

Figure 1.2 A stationary convolution process, in which a vector of seismic trace is generated by the multiplication of a wavelet matrix and a vector of reflectivity series . Each column vector of the wavelet matrix is formed by a stationary seismic wavelet .

1.2 Nonstationary Signals

The stationary convolution process from the previous section is an idealised model for seismic signals. In reality, however, seismic signals are nonstationary due to the dissipation effect when seismic waves propagate through the subsurface anelastic media.

The nonstationary convolution process can be expressed as follows,

where is a nonstationary seismic wavelet, in place of the stationary wavelet . The nonstationary seismic wavelet evolves continually according to a dissipation model:

where is the dissipation coefficient, and acts on the idealised stationary wavelet .

In the convolution expression of Eq. (1.7) for the nonstationary wavelet , the dissipation coefficient , within which indicates the time dependency, is nonstationary. At any given time position , the dissipation coefficient can be defined by

where f is the frequency, is the frequency-dependent attenuation coefficient, and is the associated dispersion, i.e. the phase delay of different frequency components (Futterman, 1962).

The attenuation coefficient of seismic waves propagating through the subsurface anelastic media is (Wang & Guo, 2004)

and the associated dispersion is

where is a reference frequency,

and is the quality factor of the subsurface anelastic media (Kolsky, 1956; Futterman, 1962; Wang, 2008). Because seismic signals have a relatively narrow frequency band, it is reasonable to assume to be frequency independent. But is time dependent, and the time here is a proxy for geologic depth.

For numerical calculation, the nonstationary convolution process of Eq. (1.6) may also be expressed in a matrix-vector form as

where is the nonstationary wavelet matrix. Figure 1.3 depicts this matrix-vector form of the nonstationary convolution process.

Figure 1.3 The nonstationary convolution process. The wavelet matrix is formed by column vectors, each of which is a nonstationary seismic wavelet , and thus the seismic trace is a nonstationary signal.

The essential distinction between stationary and nonstationary convolution models is the nonstationary wavelet matrix . While the example symmetrical wavelet in Figure 1.2 is a zero-phase wavelet, the form of the wavelet in Figure 1.3 changes continuously. The amplitude decreases and the phase varies with the travel time. The form of the wavelet gradually changes from symmetrical to asymmetrical.

In general, the differences between stationary and nonstationary signals can be captured in three characteristics.

1. Time period: The time period for a stationary signal always remains constant, whereas the time period for a nonstationary signal is not constant and varies with time.
2. Frequency: The frequency of a stationary signal remains constant throughout the process, while the frequency of a nonstationary signal changes continuously during the process.
3. Spectral content: The spectral content of a stationary signal is constant, while the spectral content of a nonstationary signal varies continuously with respect to time.

For seismic signals, there would be several sets of frequency contents within a given time interval, and these frequency contents are likely to change dynamically with respect to travel time. Therefore, the statistical properties of nonstationary seismic signals also change with time. These statistical properties include the mean, variance, and covariance.

1.3 The Fourier Transform and the Average Properties

The Fourier transform represents a stationary signal as a stack of sinusoids, each of which has a constant frequency and a constant amplitude. Therefore, the Fourier transform is a static spectral analysis that provides only the average characteristics of a nonstationary seismic signal.

For a seismic trace , the Fourier transform is defined as (Bracewell, 1965)

where is the frequency spectrum of .

The physical meaning of the Fourier transform can be understood by expressing the transform kernel in Euler's formula,

and rewriting the Fourier transform as follows,

The first integral is a cross-correlation between the seismic signal and a cosine function, , and the second integral is a cross-correlation...