# Mixed phase deconvolution

Started by March 17, 2007
```Hello all DSP gurus,
I am struggling with some (seismic) DSP theory and need some help!

In seismic the convolutional model states that the observed signal as
a function of time, st (seismic trace),
equals the convolution between the seismic shot signal, s, and the
earth response function, e:
st=s*e.
Given a known st and s I want to calculate the unknown e.
We introduce a pulse shaping filter f:
f*st=e  => f*s=dirac, where dirac is the dirac delta (1 at t=0, 0
otherwise).
In order to implement this on a computer I will have to settle for a f
of finite length and it will therefore only be an approximation (s:
FIR => f: IIR).
Therefore we choose a least squares error estimate for f and arrive at
the normal equations (why are they called this anyway?):
uss*f=usb,
uss: autocorrelation of s (matrix), f: filter vector, usb =
correlation between s and the dirac, and *: multiplication.

Now over to some data to test this!
s=[6, 5, 1] (minimum phase).
Solves the normal equations for a 3 point f:
f=[0.1588942, -0.1189128, 0.0517513]
Next I calculate the square error:
E=0.0466348
Great!

Now over to a mixed phase seismic shot signal:
s=[3, 7, 2]
Same procedure gives the following square error:
E= 0.2849761
whick sucks!

I also tried with spiking deconvolution with lag=1 and got a somewhat
better result but nowhere near the quality for the mininimum phase
signal.

So how do I go about to deconvolute my mixed phase signal with the
same level of quality as for the minimum phase signal?

Andreas Werner Paulsen

```
```On 17 Mar, 10:04, "Andreas" <andreas.werner.paul...@gmail.com> wrote:
> Hello all DSP gurus,
> I am struggling with some (seismic) DSP theory and need some help!
>
> In seismic the convolutional model states that the observed signal as
> a function of time, st (seismic trace),
> equals the convolution between the seismic shot signal, s, and the
> earth response function, e:
> st=s*e.
> Given a known st and s I want to calculate the unknown e.

I'll agree that the measured signal st is known. Are you
sure the source signature s can be assumed known? In the
days I dabbled wih such problems there were a lot of
people who did lots of work to estimate s.

> We introduce a pulse shaping filter f:
> f*st=e  => f*s=dirac, where dirac is the dirac delta (1 at t=0, 0
> otherwise).

Yes. This is a standard approach.

> In order to implement this on a computer I will have to settle for a f
> of finite length and it will therefore only be an approximation (s:
> FIR => f: IIR).

Sure. Lots of things become a lot easire for (hopefully)
only a small penalty in accuracy.

> Therefore we choose a least squares error estimate for f and arrive at
> the normal equations (why are they called this anyway?):
> uss*f=usb,

are used for system estimation where one assumes a
system excited by a stationary white noise signal.
The solution of the normal equations "re-whitens"
the signal to produce a stationary process.

In your case, the source signaure is known. I can't
see that the normal equations are the right tool for
this job.

...
> So how do I go about to deconvolute my mixed phase signal with the
> same level of quality as for the minimum phase signal?
>