# Dictionary:Complex-trace analysis

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Finding the complex number representation ${\displaystyle F(t)}$ of a real time-series ${\displaystyle f(t)}$:

${\displaystyle F(t)=f(t)+if_{\perp }(t)=A(t)e^{i\phi (t)}}$,

where ${\displaystyle f_{\perp }(t)}$ is the quadrature series, ${\displaystyle A(t)}$ is the amplitude of the envelope of the trace (also called reflection strength), and ${\displaystyle \phi (t)}$ is the instantaneous phase. Displays of instantaneous phase (or cosine of the instantaneous phase) show the continuity of an event. Instantaneous frequency is ${\displaystyle d\phi (t)/dt}$. 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.

## Mathematical foundations of complex trace analysis

The notion of a complex or analytic trace begins with the more general result that any function ${\displaystyle F(z)={\mbox{Re }}F(z)+i{\mbox{Im }}F(z)}$, where ${\displaystyle z=x+iy}$ must obey the relation that (which follows from the Cauchy integral formula) that[1] or Levinson and Redheffer (1970). [2]

${\displaystyle {\mbox{Im }}F(z)={\mathcal {H}}\left[{\mbox{Re }}F(z)\right]}$

where ${\displaystyle {\mathcal {H}}}$ is an operation known as the Hilbert transform., in any region where ${\displaystyle F(z)}$ is "analytic".("Analytic" means that ${\displaystyle dF/dz}$ exists.)

### Instantaneous amplitude

We may write (using Euler's relation)

${\displaystyle F(z)=A(z)\exp(i\phi (z))=A(z)\left[\cos(\phi (z))+i\sin(\phi (z))\right]}$

where the modulus is

${\displaystyle A(z)={\sqrt {({\mbox{Re }}F(z))^{2}+({\mbox{Im }}F(z))^{2}}}}$

.

### Instantaneous phase

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

${\displaystyle \phi (z)=\arctan \left({\frac {{\mbox{Im }}F(z)}{{\mbox{Re }}F(z)}}\right)}$.

Hence, the real part of ${\displaystyle F(z)}$ is

${\displaystyle {\mbox{Re }}F(z)=A(z)\cos(\phi (z))}$

and the imaginary part is

${\displaystyle {\mbox{Im }}F(z))=A(z)\sin(\phi (z))}$.

## The Complex (or Analytic) trace

Let us now consider a function ${\displaystyle z(t)}$ , where ${\displaystyle t}$ is monotonically increasing. The function ${\displaystyle z(t)}$ describes a curve in the ${\displaystyle (x(t),y(t),t)}$ volume. This curve is single valued in ${\displaystyle t}$ in this volume, yielding the following parameterization

${\displaystyle F(z(t))=A(z(t))\left[\cos(\phi (z(t)))+i\sin(\phi (z(t)))\right]}$

.

Because ${\displaystyle z(t)}$ is single valued in ${\displaystyle t}$ in the volume ${\displaystyle (x,y,t)}$ we can write, without loss of generality

${\displaystyle F(t)=A(t)\left[\cos(\phi (t))+i\sin(\phi (t))\right]}$

which describes a helix about the ${\displaystyle t}$axis, defined by ${\displaystyle (0,0,t)}$ in the ${\displaystyle (x,y,t)}$ volume.

Now, we assert that our recorded data ${\displaystyle f(t)}$ is the real part of this complex trace ${\displaystyle F(t)}$, hence:

${\displaystyle f(t)=A(t)\cos(\phi (t))}$

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

${\displaystyle f_{\perp }(t)=A(t)\sin(\phi (t))}$.

The modulus ${\displaystyle A(t)}$is the "instantaneous amplitude, also known as the "envelope function" of ${\displaystyle f(t)}$. The function ${\displaystyle \phi (t)}$ is known as the "instantaneous phase" of ${\displaystyle f(t)}$.

### Instantaneous frequency

An "instantaneous frequency" ${\displaystyle \omega (t)}$ may be defined as the time rate of change of the instantaneous phase ${\displaystyle \phi (t)}$

${\displaystyle \omega (t)={\frac {d\phi (t)}{dt}}}$.

.

Computationally, the instantaneous phase ${\displaystyle \phi }$ calculated in this fashion may be wrapped, which is to say it may have jumps of up to ${\displaystyle 2\pi }$ , owing to the fact that the numerical computation of the arctangent function in computers is restricted to the principle branch, ${\displaystyle -\pi <\phi \leq \pi }$.

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

${\displaystyle \omega (t)={\frac {d\phi (t)}{dt}}={\frac {d}{dt}}\arctan \left({\frac {{\mbox{Im }}F(t)}{{\mbox{Re }}F(t)}}\right)={\frac {({\mbox{Im }}F(t))^{\prime }\;({\mbox{Re }}F(t))-({\mbox{Im }}F(t))\;({\mbox{Re }}F(t))^{\prime }}{({\mbox{Re }}F(t))^{2}+({\mbox{Im }}F(t))^{2}}}}$

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

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