# vibroseis deconvolution without the target sweep

Started by October 9, 2006
I have a vibroseis dataset from which I need to find the reflection response
of the earth medium r[t].  For a simple model, assume that s[t] is the
original target sweep and s'[t] is the recorded sweep expressible by the
convolution s'[t] = s[t] * r[t].  Of course, this does not take into account
earth attenuation or noise.  What I need to find is the reflection response
r[t], but I cannot perform cross-correlation with the target sweep s[t]
because I do not have s[t].  The dataset was stored on tape, and due to
malfunction, I do not have s[t].  I also do not have the original tape,
which was destroyed a long time ago.  What can I do to find the reflection
response r[t], lacking the original sweep s[t]?

Thank you ever so much for your help!


Nicholas Kinar wrote:

> I have a vibroseis dataset from which I need to find the reflection response
> of the earth medium r[t].  For a simple model, assume that s[t] is the
> original target sweep and s'[t] is the recorded sweep expressible by the
> convolution s'[t] = s[t] * r[t].  Of course, this does not take into account
> earth attenuation or noise.  What I need to find is the reflection response
> r[t], but I cannot perform cross-correlation with the target sweep s[t]
> because I do not have s[t].  The dataset was stored on tape, and due to
> malfunction, I do not have s[t].  I also do not have the original tape,
> which was destroyed a long time ago.  What can I do to find the reflection
> response r[t], lacking the original sweep s[t]?

Nicholas,

One suggestion is to estimate the instantaneous frequency of the
vibroseis signal, call it fi_s[t], say) and form an estimate of s[t]
from

\hat{s}[t] = exp(j*sum_{v=0}^t fi_s[v])

and use this to cross-correlate with s'[t].

>From memory (and it was a long time ago, so the memory is hazy),
vibroseis signals tend to start at about 10Hz and go up to around 100Hz
[a quick google on "vibroseis" seems to confirm this memory).

Ciao,

Peter K.


Thanks, Peter.  This is very much appreciated.  Yes, indeed, vibroseis
signals are in the range that you give.  Higher frequencies tend to be more
greatly attenuated, and the reflections are weaker and thereby more
difficult to resolve.  Once again, thank you for all of your help!

Nicholas

"Peter K."  wrote in message:
> Nicholas Kinar wrote:
>
>> I have a vibroseis dataset from which I need to find the reflection
>> response
>> of the earth medium r[t].  For a simple model, assume that s[t] is the
>> original target sweep and s'[t] is the recorded sweep expressible by the
>> convolution s'[t] = s[t] * r[t].  Of course, this does not take into
>> account
>> earth attenuation or noise.  What I need to find is the reflection
>> response
>> r[t], but I cannot perform cross-correlation with the target sweep s[t]
>> because I do not have s[t].  The dataset was stored on tape, and due to
>> malfunction, I do not have s[t].  I also do not have the original tape,
>> which was destroyed a long time ago.  What can I do to find the
>> reflection
>> response r[t], lacking the original sweep s[t]?
>
> Nicholas,
>
> One suggestion is to estimate the instantaneous frequency of the
> vibroseis signal, call it fi_s[t], say) and form an estimate of s[t]
> from
>
> \hat{s}[t] = exp(j*sum_{v=0}^t fi_s[v])
>
> and use this to cross-correlate with s'[t].
>
>>From memory (and it was a long time ago, so the memory is hazy),
> vibroseis signals tend to start at about 10Hz and go up to around 100Hz
> [a quick google on "vibroseis" seems to confirm this memory).
>
> Ciao,
>
> Peter K.
>