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GCInstantaneous Previous HitSeismicNext Hit Attributes Calculated by the Hilbert Transform*

 

Bob Hardage1

 

Search and Discovery Article #40563 (2010)

Posted July 17, 2010

 

*Adapted from the Geophysical Corner column, prepared by the author, in AAPG Explorer, June, 2010, and entitled “Thin Is In: Here’s a Helpful Attribute”. Editor of Geophysical Corner is Bob A. Hardage ([email protected]). Managing Editor of AAPG Explorer is Vern Stefanic; Larry Nation is Communications Director. Please see closely related article “Reflection Events and Their Polarities Defined by the Hilbert Transform”, Search and Discovery article #40564.

 

1Bureau of Economic Geology, The University of Texas at Austin ([email protected])

 

 

General Statement

 

Geological interpretation of Previous HitseismicNext Hit data is commonly done by analyzing patterns of Previous HitseismicNext Hit amplitude, Previous HitphaseNext Hit and frequency in map and section views across a prospect area. Although many Previous HitseismicNext Hit attributes have been utilized to emphasize geologic targets and to define critical rock and fluid properties, these three simple attributes – amplitude, Previous HitphaseNext Hit and frequency – remain the mainstay of geological interpretation of Previous HitseismicNext Hit data.

 

Any procedure that extracts and displays any of these Previous HitseismicNext Hit parameters in a convenient and understandable manner is an invaluable interpretation tool. A little more than 30 years ago, M.T. Taner and Robert E. Sheriff introduced the concept of using the Hilbert transform to calculate Previous HitseismicNext Hit amplitude, Previous HitphaseNext Hit and frequency instantaneously – meaning a value for each parameter is calculated at each time sample of a Previous HitseismicNext Hit trace. That Hilbert transform approach now forms the basis by which almost all amplitude, Previous HitphaseNext Hit and frequency attributes are calculated by today’s Previous HitseismicNext Hit interpretation software

 

 

Figure Captions

 

 

General statement
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Hilbert Transform
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General statement
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Hilbert Transform
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General statement
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Hilbert Transform
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fig01

Figure 1. A complex Previous HitseismicNext Hit trace consisting of a real part x(t), which is the actual Previous HitseismicNext Hit trace, and an imaginary part y(t), which is a mathematical function calculated from the real part by a Hilbert transform. When the real and imaginary parts are added in a vector sense, the result is a helical spiral centered on the Previous HitseismicNext Hit time axis (t). This helical trace is the complex Previous HitseismicNext Hit trace.

fig02

Figure 2. Instantaneous Previous HitseismicNext Hit attributes – amplitude a(t), Previous HitphaseNext Hit Ф(t) and frequency ω(t) – that can be calculated from a complex Previous HitseismicNext Hit trace using the listed equations.

fig03

Figure 3. Instantaneous amplitude associated with Previous HitseismicNext Hit trace x1(t).

fig04

Figure 4. Instantaneous Previous HitphaseNext Hit and frequency (top two panels) associated with the Previous HitseismicNext Hit trace displayed on the bottom panel. Note that negative frequencies are calculated at times t1 and t2.

 

The Hilbert Transform

 

The action of the Hilbert transform is to convert a Previous HitseismicNext Hit trace x(t) into what first appears to be a mysterious complex Previous HitseismicNext Hit trace z(t) as shown on Figure 1. In this context, the term “complex” is used in its mathematical sense, meaning it refers to a number that has a real part and an imaginary part. The term does not imply that the data are difficult to understand. This complex trace consists of the real Previous HitseismicNext Hit trace x(t) and an imaginary Previous HitseismicNext Hit trace y(t) that is the Hilbert transform of x(t). On Figure 1 these two traces are shown in a three-dimensional data space (x, y, t), where t is Previous HitseismicNext Hit time, x is the real-data plane, and y is the imaginary-data plane. The actual Previous HitseismicNext Hit trace is confined to the real-data plane; the Hilbert transform trace is restricted to the imaginary-data plane.

 

These two traces combine to form a complex trace z(t), which appears as a helix that spirals around the time axis. The projection of complex trace z(t) onto the real plane is the actual Previous HitseismicNext Hit trace x(t); the projection of z(t) onto the imaginary plane is the Hilbert transform trace y(t). At any coordinate on the time axis, a vector a(t) can be calculated that extends perpendicularly away from the time axis to intercept the helical complex trace z(t) as shown on Figure 2. The length of this vector is the amplitude of the complex trace at that particular instant in time – hence the term “instantaneous amplitude.” The amplitude value is calculated using the equation for a(t) shown on the figure.

 

The orientation angle Ф(t) that defines where vector a(t) is pointing (Figure 2) is defined as the Previous HitseismicNext Hit Previous HitphaseNext Hit at time coordinate t – hence the term “instantaneous Previous HitphaseNext Hit.” Numerically, the Previous HitphaseNext Hit angle is calculated using the middle equation listed on Figure 2. As time progresses, vector a(t) moves down the time axis, constantly rotating about the time axis as it maintains contact with the spiraling helical trace z(t). Mathematically, frequency can be defined as the rate of change of Previous HitphaseNext Hit. This fundamental definition allows instantaneous frequency ω(t) to be calculated from the time derivative of the Previous HitphaseNext Hit function as shown by the bottom equation on Figure 2.

 

The calculation of these three interpretation attributes – amplitude, Previous HitphaseNext Hit and frequency – are illustrated on Figures 3 and 4. Application of the three equations listed on Figure 2 yields first the instantaneous amplitude for one Previous HitseismicNext Hit trace x1(t) (Figure 3), and then instantaneous Previous HitphaseNext Hit and frequency are shown on Figure 4 for a different Previous HitseismicNext Hit trace x2(t). Note that the instantaneous frequency function is occasionally negative – a concept that has great interpretation value, as has been discussed in a previous article (Interpretation Value of Anomalous (‘Impossible’) Frequencies, Search and Discovery article #40286). For those of you who click on a menu choice to create a Previous HitseismicNext Hit attribute as you interpret Previous HitseismicNext Hit data, you now see what goes on behind the screen to create that attribute.

 

Reference

 

Taner, M.T. and Robert E. Sheriff, 1977, Application of Amplitude, Frequency, and Other Attributes to Stratigraphic and Hydrocarbon Determination: Section 2. Application of Previous HitSeismicTop Reflection Configuration to Stratigraphic Interpretation, AAPG Memoir 26, p. 301-327.

 

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