Dictionary:Complex-trace analysis

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

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

where $f_{\perp }(t)$ is the quadrature series, $A(t)$ is the amplitude of the envelope of the trace (also called reflection strength), and $\phi (t)$ is the instantaneous phase. Displays of instantaneous phase (or cosine of the instantaneous phase) show the continuity of an event. Instantaneous frequency is $d\phi (t)/dt$. 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 $F(z)={\mbox{Re }}F(z)+i{\mbox{Im }}F(z)$, where $z=x+iy$ must obey the relation that (which follows from the Cauchy integral formula) that or Levinson and Redheffer (1970).

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

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

Instantaneous amplitude

We may write (using Euler's relation)

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

where the modulus is

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

.

Instantaneous phase

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

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

Hence, the real part of $F(z)$ is

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

and the imaginary part is

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

The Complex (or Analytic) trace

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

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

.

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

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

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

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

$f(t)=A(t)\cos(\phi (t))$

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

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

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

Instantaneous frequency

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

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

.

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

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

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

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) , has found wide application in seismic processing for interpretation.

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

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