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# Seismic attributes

The fundamental properties of processed seismic data that are used in interpretation are temporal and spatial variations of:

• Reflection amplitude
• Reflection phase
• Wavelet frequency

Structural and stratigraphic interpretations of 3D seismic data are inferences of geologic conditions made by analyzing areal patterns of these three seismic attributes across selected seismic time surfaces:

• Amplitude
• Phase
• Frequency

Any procedure that extracts and displays these seismic parameters in a convenient, understandable format is an invaluable interpretation tool.

Taner and Sheriff and Taner et al. began using the Hilbert transform to calculate seismic amplitude, phase, and frequency instantaneously, meaning that a value of amplitude, phase, and frequency is calculated for each time sample of a seismic trace. Since that introduction, numerous Hilbert transform algorithms have been implemented to calculate these useful seismic attributes (e.g., Hardage).

## Complex seismic trace

Fig. 1 illustrates the concept of a complex seismic trace in which x(t) represents the real seismic trace and y(t) is the Hilbert transform of x(t). In this discussion, we ignore what a Hilbert transform is and how the function y(t) is calculated. Most modern seismic data-processing software packages provide Hilbert transform algorithms and allow processors to create the function y(t) shown in Fig. 2 easily. These two data vectors are displayed in a 3D (x, y, t) space in which t is seismic traveltime, x is the real data plane, and y is the imaginary plane. In this complex trace format, the actual seismic trace, x(t), is confined to the real x plane, and y(t), the Hilbert transform of x(t), is confined to the imaginary y plane. When x(t) and y(t) are added vectorally, the result is a complex seismic trace, z(t), in the shape of a helical spiral extending along, and centered about, the time axis t. The projection of this complex function z(t) onto the real plane is the real seismic trace, x(t), and the projection of z(t) onto the imaginary plane is y(t), the calculated Hilbert transform of x(t).

Fig. 2 illustrates the reason for converting the real seismic trace, x(t), into what first appears to be a more mysterious complex seismic trace, z(t), in which the attributes known as instantaneous seismic amplitude, instantaneous phase, and instantaneous frequency are introduced. At any point on the time axis of this complex seismic trace, a vector a (t) can be calculated that extends away from the t axis in a perpendicular plane to intersect the helically shaped complex seismic trace, z(t). The length of this vector is the amplitude of the complex trace at that particular instant, hence the term instantaneous amplitude. This amplitude value is calculated with the equation for a (t) shown in Fig. 2.

The orientation angle, ϕ(t), of the amplitude vector, a(t), at time t, which is generally measured relative to the positive axis of the real x-plane, is defined as the phase of z(t) at that moment in time. Numerically, the phase angle is calculated from the equation for ϕ(t) defined in Fig. 2. As seismic time progresses, vector a(t) moves along the time axis and rotates continually about the time axis to maintain contact with the spiraling complex trace, z(t). Each full rotation of the vector around the time axis increases the phase value by 360°.

In any oscillating system, and specifically for a seismic trace, frequency can be defined as the time rate of change of the phase angle. This fundamental definition describes the frequency of the complex seismic trace so that the instantaneous frequency, ω(t), at any seismic time sample is given by the time derivative of the phase function specified by the equation in Fig. 2.

## Instantaneous phase and instantaneous frequency calculations

Fig. 3 illustrates calculations of the instantaneous phase associated with a typical seismic trace. The figure’s bottom panel shows the actual seismic trace, and the center panel shows the real and imaginary components of the associated complex trace. Applying the second equation of Fig. 4 to the real and imaginary components of the complex seismic trace (center panel) produces the instantaneous phase function at the top of Fig. 3. The phase behavior at times t1, t2, t3 is critical to understanding the geologic significance of anomalous frequencies. Although phase is a positive function that monotonically increases in magnitude with seismic time, it is customarily plotted as a repetitive, wraparound function with plot limits of 0 to 360° (or –180° to +180°). Each wraparound of 360° corresponds to a full rotation of vector a(t) around the seismic time axis while the vector stays in contact with the spiraling complex seismic trace z(t) (see Fig. 4).

The top panel of Fig. 5 shows the instantaneous frequencies calculated for the seismic trace presented in Fig. 3. Although the calculated frequency values at times t1 and t2 shown by the solid-line curve are physically impossible because they are negative, these anomalous frequency behaviors are some of the most useful seismic attributes that an interpreter can use. They should be preserved because, when displayed in an eye-catching, contrasting color, they serve to shift an interpreter’s attention quickly to subtle structural and stratigraphic discontinuities in a seismic image.

Comparing the time coordinates of these anomalous frequency values with the time coordinates of the instantaneous phase function in Fig. 3 shows that rather than exhibiting its typical, monotonically increasing behavior in these time intervals, the phase momentarily decreases in magnitude. This causes the time rate of change of phase (or the slope of the phase function), which is the instantaneous frequency, to be negative at time samples t1 and t2. Numerical algorithms should not camouflage these unrealistic frequency values, which some algorithms do by arbitrarily reversing the algebraic sign of any negative frequency. The dashed-curve segments in the enlargement show how some software algorithms change the algebraic sign of negative frequency values to positive values. Although this seems like a logical correction, it should be avoided because it reduces the interpretive value of a 3D volume of instantaneous frequencies.

At times t1, t2, and t3, some type of wavelet interference (that is, a wavelet distortion) occurs in the seismic trace (bottom panel). As a result, the reflection waveform at times t1, t2, and t3 is slightly distorted (bottom panel) because of the destructive interference of two or more overlapping wavelets, demonstrating that anomalous frequencies tend to coincide with, and emphasize, distorted wavelets such as are produced at structural and stratigraphic discontinuities. High positive frequency values also occur at 1580, 1650, 1740, and 1840 milliseconds, but these values are not anomalous in the sense that they do not exceed the Nyquist limit (125 Hz). However, they too coincide with distorted wavelets (bottom panel).