# Wiener filters for transient deconvolution

Started by November 1, 2004
```Hi folks.

I'm sitting here with some data recorded by a sonar. The sonar is
directed vertically, insonifying a layered sea floor. My job is to
try to estimate the reflection series for this sea floor.
I have both a nominal source wavelet (i.e. the analytical expression
for the emitted pulse) and a reference recording, where the sonar was
directed towards a reciever that was suspended in water.

Now, I'm trying to use a Wiener filter for the deconvolution of the
recorded signal. The expression for the Wiener filter involves the
covariance matrixes for both signal and noise. There is an interval
between the emitted pulse and the first recieved echo, that I can use
as 'noise only' reference data. The source wavelet is more or less
known, so I possess very priveledged information, compared to what I'm
used to from seismics.

Now I would like to take advantage of this gold mine of information,
but I'm facing a few difficulties in doing so.

First of all, my idea behind using a Wiener filter is that the basic
expressions aim at compressing the cross covariance R_xd between
the measured signal x and the desired signal d (= wavelet transient)
into a delta series, i.e. producing a spike. A matched filter, on
the other hand, aims at maximizing the correlation measure between
x and d, which is a completely different thing. The guys who did
the survey, used a frequency sweep pulse in parts of the survey,
where a matched filter might work better. Unfortunately, the bulk of
the survey (including the most interesting parts) was done using a
Ricker wavelet.

Does anybody have any comments on this argument for choosing a
Wiener filter? Am I right in my comparison between the Wiener filter
and the matched filter, or did I misunderstand something?

Second, Wiener filters are defined for stationary time series, i.e.
non-transient, 'noise-like' data. My best shot here, is to estimate
a low-order autocovariance matrix R_dd for the reference wavelet data,
and use that with the estimated noise-only covariance matrix R_nn to
estimate the filter coefficients from the equation

(R_dd+R_nn)h = r_dx = [sigma^2, 0, 0, ...]^T.

Have anybody tried to do this for transients? Any thoughts? Pitfalls?

Rune
```
```This is almost ridiculous.

After posting yesterday, I did some searches in the seismic literature
for "predictive deconvolution", which is a standard tool in seismics.
It turns out there was a lot of activity in the 1970ies-80ies on this
problem, and some solutions even included Wiener filters. I have yet
to find any mentioning of a Matched Filter, but that might have just
as much to do with that source wavelets are generally not known in
seismic problems, as anything else.

The problem is that it's easy to find mentioning of "predictive decon"
and no seismic processing package comes without this operator.
Everybody in the business knows what it is supposed to do (separate
a recorded data series into a source wavelet and a reflection sequence),
and the rudimentaries of how it works ("Inverse filter", "Wiener filter").
But for some reason it seems to be very difficult, if not impossible,
to find out the detailed algorithms.

Oh well. At least I found some of the right key words. I am not
completely lost.

Rune

allnor@tele.ntnu.no (Rune Allnor) wrote in message news:<f56893ae.0411010403.41fe6c92@posting.google.com>...
> Hi folks.
>
> I'm sitting here with some data recorded by a sonar. The sonar is
> directed vertically, insonifying a layered sea floor. My job is to
> try to estimate the reflection series for this sea floor.
> I have both a nominal source wavelet (i.e. the analytical expression
> for the emitted pulse) and a reference recording, where the sonar was
> directed towards a reciever that was suspended in water.
>
> Now, I'm trying to use a Wiener filter for the deconvolution of the
> recorded signal. The expression for the Wiener filter involves the
> covariance matrixes for both signal and noise. There is an interval
> between the emitted pulse and the first recieved echo, that I can use
> as 'noise only' reference data. The source wavelet is more or less
> known, so I possess very priveledged information, compared to what I'm
> used to from seismics.
>
> Now I would like to take advantage of this gold mine of information,
> but I'm facing a few difficulties in doing so.
>
> First of all, my idea behind using a Wiener filter is that the basic
> expressions aim at compressing the cross covariance R_xd between
> the measured signal x and the desired signal d (= wavelet transient)
> into a delta series, i.e. producing a spike. A matched filter, on
> the other hand, aims at maximizing the correlation measure between
> x and d, which is a completely different thing. The guys who did
> the survey, used a frequency sweep pulse in parts of the survey,
> where a matched filter might work better. Unfortunately, the bulk of
> the survey (including the most interesting parts) was done using a
> Ricker wavelet.
>
> Does anybody have any comments on this argument for choosing a
> Wiener filter? Am I right in my comparison between the Wiener filter
> and the matched filter, or did I misunderstand something?
>
> Second, Wiener filters are defined for stationary time series, i.e.
> non-transient, 'noise-like' data. My best shot here, is to estimate
> a low-order autocovariance matrix R_dd for the reference wavelet data,
> and use that with the estimated noise-only covariance matrix R_nn to
> estimate the filter coefficients from the equation
>
>   (R_dd+R_nn)h = r_dx = [sigma^2, 0, 0, ...]^T.
>
> Have anybody tried to do this for transients? Any thoughts? Pitfalls?
>
> Rune
```
```Hi Rune,

My 2 cents...

I'm a little confused at what the goal is. Is the goal to
a) Detect and measure the delay between the transmitted pulse and the
echo(es), or;
b) To find the optimal filter that models the environment

If the goal is (a), my understanding is that the matched filter will
provide the likelihood function measure that you can use for a
hypothesis test to detect the presence/absence of a pulse (and thereby
measure the delay). On the other hand, if the goal is (b) then the
Weiner filter will provide the filter coefficients that will minimize
the mean squared error (that is, it would try to make (X-Xd) orthogonal
to your observation space {Y} => E[(X-Xd)Y]=0).

- Ravi

Rune Allnor wrote:
> Hi folks.
>
> I'm sitting here with some data recorded by a sonar. The sonar is
> directed vertically, insonifying a layered sea floor. My job is to
> try to estimate the reflection series for this sea floor.
> I have both a nominal source wavelet (i.e. the analytical expression
> for the emitted pulse) and a reference recording, where the sonar was
> directed towards a reciever that was suspended in water.
>
> Now, I'm trying to use a Wiener filter for the deconvolution of the
> recorded signal. The expression for the Wiener filter involves the
> covariance matrixes for both signal and noise. There is an interval
> between the emitted pulse and the first recieved echo, that I can use
> as 'noise only' reference data. The source wavelet is more or less
> known, so I possess very priveledged information, compared to what I'm
> used to from seismics.
>
> Now I would like to take advantage of this gold mine of information,
> but I'm facing a few difficulties in doing so.
>
> First of all, my idea behind using a Wiener filter is that the basic
> expressions aim at compressing the cross covariance R_xd between
> the measured signal x and the desired signal d (= wavelet transient)
> into a delta series, i.e. producing a spike. A matched filter, on
> the other hand, aims at maximizing the correlation measure between
> x and d, which is a completely different thing. The guys who did
> the survey, used a frequency sweep pulse in parts of the survey,
> where a matched filter might work better. Unfortunately, the bulk of
> the survey (including the most interesting parts) was done using a
> Ricker wavelet.
>
> Does anybody have any comments on this argument for choosing a
> Wiener filter? Am I right in my comparison between the Wiener filter
> and the matched filter, or did I misunderstand something?
>
> Second, Wiener filters are defined for stationary time series, i.e.
> non-transient, 'noise-like' data. My best shot here, is to estimate
> a low-order autocovariance matrix R_dd for the reference wavelet data,
> and use that with the estimated noise-only covariance matrix R_nn to
> estimate the filter coefficients from the equation
>
>   (R_dd+R_nn)h = r_dx = [sigma^2, 0, 0, ...]^T.
>
> Have anybody tried to do this for transients? Any thoughts? Pitfalls?