# Dictionary:Complex-trace analysis

Finding the complex number representation **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(t)}**
of a real time-series **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(t)}**
:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(t)= f(t)+ i f_{\perp}(t)=A(t)e^{i \phi(t)} }**,

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{\perp}(t) }**
is the **quadrature series**, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A(t)}**
is the **amplitude of the envelope** of the trace (also called **reflection strength**), and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi(t)}**
is the **instantaneous phase**. Displays of instantaneous phase (or cosine of the instantaneous phase) show the continuity of an event. **Instantaneous frequency** is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d\phi(t)/dt}**
. Instantaneous frequency can be thought of as the frequency of the complex sinusoid that locally best fits a complex trace. Used to determine seismic attributes. In the space domain, "local" is sometimes used instead of "instantaneous". See Figure C-11 y Taner *et al.* (1979). Complex-trace analysis often involves the Hilbert transform.

## Contents

## Mathematical foundations of complex trace analysis

The notion of a complex or analytic trace begins with the more general result
that any function **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(z)= \mbox{Re } F(z) + i \mbox{Im } F(z) }**
, where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z = x + i y }**
must obey the relation that
(which follows from the Cauchy integral formula) that^{[1]} or Levinson and Redheffer (1970). ^{[2]}

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mbox{Im } F(z) = {\mathcal H} \left[ \mbox{Re }F(z) \right] }**

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\mathcal H} }**
is an operation known as the *Hilbert transform*., in any region
where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(z) }**
is "analytic".("Analytic" means that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle dF/dz }**
exists.)

### Instantaneous amplitude

We may write (using Euler's relation)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(z) = A(z)\exp(i \phi(z)) = A(z)\left[ \cos(\phi(z) ) + i \sin(\phi(z)) \right]}**

where the modulus is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A(z) = \sqrt{ (\mbox{Re } F(z))^2 + (\mbox{Im } F(z))^2 } }**

.

### Instantaneous phase

The phase, then is the arc tangent of the ration of the imaginary and real parts

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi(z) = \arctan \left( \frac{\mbox{Im } F(z)}{\mbox{Re } F(z)} \right) }**.

Hence, the real part of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(z) }**
is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mbox{Re } F(z) = A(z) \cos(\phi(z)) }**

and the imaginary part is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mbox{Im } F(z)) = A(z) \sin(\phi(z))}**.

## The Complex (or Analytic) trace

Let us now consider a function **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z(t)}**
, where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t}**
is monotonically
increasing. The function **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z(t)}**
describes a curve in the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (x(t) ,y(t) ,t)}**
volume.
This curve is single valued in **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t}**
in this volume, yielding the following
parameterization

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(z(t)) = A(z(t)) \left[ \cos(\phi(z(t))) + i \sin(\phi(z(t))) \right] }**

.

Because **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z(t) }**
is single valued in **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t }**
in the volume **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (x,y,t)}**
we can
write, without loss of generality

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(t) = A(t)\left[\cos (\phi(t)) + i \sin(\phi(t)) \right] }**

which describes a helix about the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t }**
axis, defined by **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (0,0,t)}**
in the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (x,y,t)}**
volume.

Now, we assert that our recorded data **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(t)}**
is the real part of this complex
trace **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(t)}**
, hence:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(t) = A(t) \cos (\phi(t)) }**

and the imaginary part, or the so-called "quadrature trace" is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{\perp} (t) = A(t) \sin (\phi(t)) }**.

The modulus **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A(t)}**
is the "instantaneous amplitude,* also known as the "envelope function" of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(t) }
.*
The function

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi(t)}**is known as the "instantaneous phase" of

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(t)}**.

### Instantaneous frequency

An "instantaneous frequency" **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega(t)}**
may be defined as the time rate of change of the instantaneous phase **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi(t)}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega(t) = \frac{d\phi(t)}{dt} }**.

.

Computationally, the instantaneous phase **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi }**
calculated in this fashion may be *wrapped,* which
is to say it may have jumps of up to **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2 \pi }**
, owing to the fact that the
numerical computation of the arctangent function in computers is restricted to the principle branch, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -\pi < \phi \le \pi }**
.

It is preferable to differentiate the arctangent function, itself, to avoid phase wrapping issues

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega(t) = \frac{d\phi(t)}{dt} = \frac{d}{dt} \arctan \left( \frac{\mbox{Im } F(t)}{\mbox{Re } F(t)} \right) = \frac{ (\mbox{Im } F(t))^\prime \; (\mbox{Re } F(t)) - (\mbox{Im } F(t)) \; (\mbox{Re } F(t)) ^\prime}{(\mbox{Re } F(t))^2 + (\mbox{Im } F(t))^2} }**

where we recognize that the denominator is the instantaneous amplitude squared.

This formulation of complex trace analysis, introduced into the geophysical community by Taner, Koehler, and Sheriff (1979) ^{[3]}, has found wide application in seismic processing for interpretation.

Since 1979, a collection of so-called seismic trace attributes have been created.

## References

- ↑ Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).
- ↑ Levinson, Norman, and Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.
- ↑ M. T. Taner, F. Koehler, and R. E. Sheriff (1979). ”Complex seismic trace analysis.” GEOPHYSICS, 44(6), 1041-1063. doi: 10.1190/1.1440994