# DC component: Seismic inversion

Multiple papers (for example, Limitations on impedance inversion of band-limited reflection data Gosh[2000]) and textbooks covering seismic inversion mention a necessary component of any seismic inversion: the DC (or zero frequency) component. For lack of a better question, what does a "zero frequency" component look like? Also, why is this considered a necessary component of any complete seismic inversion scheme?

The impedance inversion problem asks: "Recover acoustic impedance from seismic reflection waveforms". The forward modeling procedure from acoustic impedance to seismic reflection waveforms involves calculating reflection coefficients and then convolving these with a wavelet.

Let's assume, for now, that you have perfect seismic resolution and therefore your wavelet is of infinite bandwidth. This means that the inversion problem becomes: "Recover the acoustic impedance from a given set of reflection coefficients".

As shown by Peterson et al. (1955), for small contrasts in acoustic impedance, the reflection coefficient equation can be linearized as shown below. Linear problems are much much easier to solve computationally than non-linear. This makes the forward model calculation a simple difference, the discrete analog to a derivative. Note: this is for normal incidence but a similar analysis can be done for non-normal incidence.

So... if it takes a derivative to calculate the reflection coefficient, it must take an integral to calculate the acoustic impedance. However, as everyone knows, when you integrate any function, you inevitably run into the non-uniqueness caused by that dreaded +C at the end of all integrals.

That +C, in this context, means that there are an infinite number of acoustic impedance models that can give you the exact same reflection coefficients. So you need the +C (i.e. the DC, or zero frequency, or additive constant) information to do a complete inversion and really recover the true values of impedance in the earth. All you get from the seismic are the reflection coefficients (and that's with infinite resolution), so you will need to get this DC information from another independent source.

In practice, the inversion is not done by a simple integration but by setting up a linear system and iteratively converging (by something like conjugate gradient) to the answer that minimizes some misfit function between the forward model and the data. As complex as that sounds, deep down, we are pretty much just doing a careful integration.

Hope that helped.

Peterson, R. A., W. R. Fillippone, and F. B. Coker, 1955, The synthesis of seismograms from well log data: Geophysics, 20 (3), 516–538.