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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