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Book Chapter

Phase

Published:
January 01, 2008

Abstract

Chapter 12 introduced the concept of instantaneous attributes. Instantaneous means that we consider properties that are associated with the trace at one instant of discrete time n. A trace (which is a real-valued signal) can be considered to be the real part of a complex trace 
formula
In this representation, the trace xn can be called the in-phase signal, and the imaginary part yn is called the quadrature signal. The quadrature signal is the Hilbert transform of the in-phase signal. The complex trace can be written as a vector, which in polar form is 
formula
where 
formula

The length An is called the instantaneous amplitude, and the angle θn is called the instantaneous phase lead. For the rest of this chapter, we shall simply say phase instead of phase lead.

What is phase rotation? Phase rotation refers to rotation of the complex-trace vector. Phase rotation for a specified phase angle is done as follows: A rotation of 90° (π/2 radians) transforms the complex trace, as given by equation 2 of Chapter 12, into the rotated complex trace 
formula

We recognize the real part, namely un = −An sin θn, as the negative of the quadrature component (Figure 1).

Figure 1.

A complex-trace z and the rotated complex trace w for a rotation angle of 90°.

Figure 1.

A complex-trace z and the rotated complex trace w for a rotation angle of 90°.

A 180° rotation gives the factor e = −1. Thus, the rotated complex trace is

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Contents

Society of Exploration Geophysicists Geophysical References Series

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

Enders A. Robinson
Enders A. Robinson
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Sven Treitel
Sven Treitel
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Society of Exploration Geophysicists
Volume
15
ISBN electronic:
9781560801610
Publication date:
January 01, 2008

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