Mixed phase deconvolution

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?

Thaks in advance for any answers!
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,

Eh... I am not sure about this. The normal equations
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?
>
> Thaks in advance for any answers!
> Andreas Werner Paulsen

Assuming that the source signature really is known,
I'd try the matched filter. It ought to work provided
you really do know the source signature. However, for
seismic field data, this might be a somewhat optimistic
assumption.

Rune