simultaneous frequence and phase estimation

christensen@nospam.ieee.org (Mads G. Christensen) wrote in message news:<wkyisfsfm53.fsf@zil.kom.auc.dk>...
> Hi Rune.
> 
> Rune> A more interesting observation is that papers that describe
> Rune> these types of methods applied to real-world data are simply
> Rune> missing from the journals.  I have yet to see any article
> Rune> describing any frequency estimator being applied to real data in
> Rune> real environments in "production type" applications. Heck, I
> Rune> don't even know of any published real-world lab tests (as
> Rune> opposed to computer-generated synthetic data) with these types
> Rune> of techniques.
> 
> Rune> I find this deafening silence with respect to practical
> Rune> applications and real-world results quite worrying.
> 
> Well, ESPRIT has been used (and you could very well use MUSIC, I
> believe) for sinusoidal modeling and coding of speech and audio, and
> there are papers on this in IEEE Trans. on Speech and Audio
> Processing. Since different applications have different journals, no
> wonder that one does not encounter these in the more theoretical
> journals such as IEEE Trans. on Signal Processing. They don't belong
> there. From your post I take it, that these methods are not applied in
> your field. That doesn't mean that there aren't applications, though.

"Have been used" is a vague term. I am interested in seeing these types 
of methods being used in a running environment where there are 
requirements to fail-safe processing and robust performance.

I ran a search on www.ieee.org/ieeexplore,

('music algorithm' <or> 'esprit algorithm') <and> (application <or> applied)

and found a total of 55 articles, of which prehaps five treated these 
methods applied to real-world problems or data. Assuming MUSIC has been 
around for 25 years and ESPRIT for 15, five articles describing 
applications is not a lot. That's one article every five years or so.

I have used frequency estimators to solve practical problems, although 
I have not published any of my results in the journals. During my work 
with the frequency estimators, I found that there are several reasons 
why they would not work, there are several problems to get them to work, 
and one must be very, very careful in interpreting and using the 
results. The problem is that I see no indications of the authors 
of research articles and textbooks being aware of the same problems 
and subtleties that I learned of in the hard way. 
 
Rune

Hi Rune.

Rune> "Have been used" is a vague term. I am interested in seeing
Rune> these types of methods being used in a running environment where
Rune> there are requirements to fail-safe processing and robust
Rune> performance.

Sure, they have been used, and papers on these applications exist. You
were saying that you couldn't find any papers documenting
applications. By applications I though you meant applied as in
used. I don't understand why you find this vague.

That the papers may not include the kind of evaluation you are looking
for is probably due to other reasons such as the complexity of the
experiments. At least that is the case in speech and audio
processing where the perceived quality (determined by listening tests)
is usually the measure of performance. 

Rune> I ran a search on www.ieee.org/ieeexplore,
Rune> ('music algorithm' <or> 'esprit algorithm') <and> (application
Rune> <or> applied)
Rune> and found a total of 55 articles, of which prehaps five treated
Rune> these methods applied to real-world problems or data. Assuming
Rune> MUSIC has been around for 25 years and ESPRIT for 15, five
Rune> articles describing applications is not a lot. That's one
Rune> article every five years or so.

For audio coding applications (my field), for example, the
computational complexity of these algorithms compared to the
alternatives may be prohibitive. The difficulties in incorporating
perception may be another reason for not using them (although it has
been done for ESPRIT). Anyways, my point is just that there may be
many reasons for the lack of papers.

I think that in many areas, the subspace methods are still considered
kind of excotic.

MUSIC also seems to be fairly popular in estimation problems in
communication. At least I know people who use it in such
applications. Again, I think that computational complexity is the
reason that there haven't been many papers in the past on applications
in that area. 

Rune> I have used frequency estimators to solve practical problems,
Rune> although I have not published any of my results in the
Rune> journals. During my work with the frequency estimators, I found
Rune> that there are several reasons why they would not work, there
Rune> are several problems to get them to work, and one must be very,
Rune> very careful in interpreting and using the results. The problem
Rune> is that I see no indications of the authors of research articles
Rune> and textbooks being aware of the same problems and subtleties
Rune> that I learned of in the hard way.

Are you thinking of, for example, spurious estimates, clustered
estimates for transient signals? What algorithm doesn't have such
problems? That the fundamental assumptions behind some solution may
turn out to be invalid in some applications is not specific to these
algorithms. I agree that you have to be careful in interpreting the
results, but that goes for any frequency estimator I know of. Anyways,
if your point is simply that you have to know what you are doing and
that these algorithms can be cumbersome, then I agree.

My guess is that the reason that you don't see many papers concerning
the problems in applications is, that tractable experiments showing
these kinds of problems are not very easily made and evaluated in a
formal way. Besides, who publishes negative results (although negative
results may prevent others from making the same mistakes)? :-)

-- 
/Mads (http://kom.aau.dk/~mgc)

Michal Kvasnicka wrote:
> Hi Paul,
> 
> I am looking for adaptive high-resolution methods. FFT is standard but not
> always suitable for this type of problems.
> 
> Michal (ERA)
> "Paul Howland" <howland@wanadoo.nl> p&#4294967295;se v diskusn&#4294967295;m pr&#4294967295;spevku

Sounds like an Eastern European language. From dsp, radar, and matlab I 
infer that you are perhaps not dealing with audio technology being 
close to my field of knowledge but possibly with spread spectrum 
experiments.
In that case, temporal structure plays a role, too. So you might 
nevertheless be interested in my 'natural spectrogram: no arbitrary 
window, no trade-off' even if FCT does not at all provide phase.

Very nice...
Could you be more specific and less talkative? What exactly is "natural
spectrogram"?

Michal
"Eckard Blumschein" <blumschein@et.uni-magdeburg.de> p&#4294967295;se v diskusn&#4294967295;m
pr&#4294967295;spevku news:4087E522.8040106@et.uni-magdeburg.de...
>
>
> Michal Kvasnicka wrote:
> > Hi Paul,
> >
> > I am looking for adaptive high-resolution methods. FFT is standard but
not
> > always suitable for this type of problems.
> >
> > Michal (ERA)
> > "Paul Howland" <howland@wanadoo.nl> p&#4294967295;se v diskusn&#4294967295;m pr&#4294967295;spevku
>
> Sounds like an Eastern European language. From dsp, radar, and matlab I
> infer that you are perhaps not dealing with audio technology being
> close to my field of knowledge but possibly with spread spectrum
> experiments.
> In that case, temporal structure plays a role, too. So you might
> nevertheless be interested in my 'natural spectrogram: no arbitrary
> window, no trade-off' even if FCT does not at all provide phase.
>
>
>
>
>
>
>

christensen@nospam.ieee.org (Mads G. Christensen) wrote in message news:<wkyzn94dq5s.fsf@zil.kom.auc.dk>...
> Hi Rune.

> Sure, they have been used, and papers on these applications exist. You
> were saying that you couldn't find any papers documenting
> applications. By applications I though you meant applied as in
> used. I don't understand why you find this vague.

There are several meanings here. "I have implemented MUSIC as a homework 
computer assignment in class" is one, "I have designed a special 
variation of MUSIC that take advantage of some exotic condition hardly 
ever found in practice" is another, "I tested MUSIC with syntentic 
signals with different SNRs" is yet another. The use I don't see is 
as in "I used MUSIC in an unsupervised analysis system,applied to 
real-world measurements in production applications".
 
> That the papers may not include the kind of evaluation you are looking
> for is probably due to other reasons such as the complexity of the
> experiments. 

Sure. That's why I haven't published anything. I find it not worth the 
while to squeeze al relevant background, examples and analysis etc into
10 or 12 pages, which usually is the budget of an article.

 > For audio coding applications (my field), for example, the
> computational complexity of these algorithms compared to the
> alternatives may be prohibitive. The difficulties in incorporating
> perception may be another reason for not using them (although it has
> been done for ESPRIT). Anyways, my point is just that there may be
> many reasons for the lack of papers.

My point exactly. I found several problematic aspects of these 
types of methods when I worked with them. The authors of journal 
articles appear not to be aware of the problems.
 
> I think that in many areas, the subspace methods are still considered
> kind of excotic.

That's because they are. Unfortunately, one have to be very familiar 
with both linear algebra and vector spaces to work with these types 
of methods. Otherwise, they fail. Big time. That happens even with 
people who do know linear algebra and vector spaces...

> MUSIC also seems to be fairly popular in estimation problems in
> communication. At least I know people who use it in such
> applications. Again, I think that computational complexity is the
> reason that there haven't been many papers in the past on applications
> in that area. 

Believe me, numerical complexity is the least problem with MUSIC.
 
> Are you thinking of, for example, spurious estimates, clustered
> estimates for transient signals? What algorithm doesn't have such
> problems? That the fundamental assumptions behind some solution may
> turn out to be invalid in some applications is not specific to these
> algorithms. 

Perhaps not specific but such assumptions are qruical in getting 
half-relevant results. Try, for instance, to implement MUSIC with 
a signal covariance matrix of order 8. Then generate a synthetic 
signal, SNR of your choosing, that comprises 10 complex exponentials
at arbitrary frequencies. How does MUSIC perform? What indication 
do you see that MUSIC has broken down? What will the cost of 
such a break-down be in a real-world applications?

> I agree that you have to be careful in interpreting the
> results, but that goes for any frequency estimator I know of. 

Of course. I mention MUSIC since that's the one everybody knows. 

> Anyways,
> if your point is simply that you have to know what you are doing and
> that these algorithms can be cumbersome, then I agree.

I would go a bit further. These algorithms are so cumbersome that 
one should not expect them to work at all. Now, that applies to a 
certain extent to class-room (or lab) excercises with synthetic 
data, but the huge problems appear when you try to take the methods 
out of the computer lab and try to use the methods with real data. 

> My guess is that the reason that you don't see many papers concerning
> the problems in applications is, that tractable experiments showing
> these kinds of problems are not very easily made and evaluated in a
> formal way. Besides, who publishes negative results (although negative
> results may prevent others from making the same mistakes)? :-)

Exactly. So we agree one should expect some sort of bias in the 
literature. It appears we agree on most points. 

Rune

Michal Kvasnicka wrote:
> Very nice...
> Could you be more specific and less talkative? What exactly is "natural
> spectrogram"?

Traditional spectrograms are based on complex-valued Fourier transform 
but they do not directly and not completely represent phase information. 
They omit phase because the ears are believed to ignore phase anyway 
which is not exactly true. Exhibiting further serious flaws, the usual 
spectrograms are is not a proper basis for a realistic description of 
cochlear function.

There is only one natural spectrogram. It is based on real-valued 
Fourier cosine transform which does not provide magnitude vs. phase but 
amplitude as a function of frequency and time, instead. So the 
information is conveyed without any loss, and in a physiologically 
realistic manner. Amplitudes alternate, so they may be rectified. The 
natural spectrogram shows largely the same output as does basilar membrane.

While such physiological details are not of interest here, I would like 
to stress a fundamental peculiarity. Use of FCT implies the possibility 
to chose a sliding point of time reference. If one intends to avoid 
permanent relocation of the time window for real-time analysis then 
there is no alternative but to use this admittedly uncommon notion of 
elapsed time instead of usual time. Actually elapsed time is always zero 
at the border towards future. Future does not matter in physics.

You are certainly aware that it is strictly speaking physical nonsense 
to integrate over time from minus infinite to plus infinite. Causality 
limits relevant time to the past one. So Heaviside's trick of adding 
mutually cancelling even and odd mirror pictures into the empty half 
plane is a bit self-deceptive. It necessarily causes Hermitian 
redundancy. Negative values of frequency or wave number are due to the 
neglect of the half of rotating phasors in complex plane and do not 
correspond to any physical meaning.

Real-valued frequency analysis is furthermore natural in that, it avoids 
a lot of arbitrariness. Already the choice of time reference t=0 at the 
natural border towards past is the only available natural one, while 
complex FT suffers from linear phase and is based on the neglect of 
either arbitrarily excluded clockwise or anti-clockwise rotating 
phasors. One arbitrariness demands an other one. In the end, users of a 
spectrogram have to choose shape, width, repetition rate, or overlap, 
respectively of the temporal window. FT is even to blame for the 
notorious dielemma between spectral and temporal resolution. Not so with 
FCT.

Do not worry about FCT-mathematics. DCT works well in MP3. FCT may be 
combined with subsequent complex calculus in order to completely replace 
FT, if necessary. Of course, FCT is not a tool for deterministically 
predicting a tree of responses but it is adequate to the logically 
inversely structured task of signal analysis. FCT-mathematics offers 
nice simplifications. Not just the number of so called singularity 
functions is considerably reduced. More importantly, the remaining 
constitute a naturally ordered system, and integration constants are no 
longer necessary. No information is lost with inversion of integration.

Admittedly, I am facing trouble with those who have to accept some 
amendments. I do not preach against negative values of frequency or wave 
number. Constituting inavoidable redundancy they are justified with FT. 
However, I would like to clearly point out that Hermitian symmetry is a 
must if the underlying quantity obeys unilaterality as do elapsed time 
and radius. This limits validity of the shift theorems.

Quantum physicist meanwhile avoid discussing possible consequences. 
Maybe, they dislike the ideas that elapsed time has a natural zero 
because they do not yet consider Hermitian PCT symmetries just due to 
complex analysis. Introducing wave function immediately within complex 
plane, they could not even tell me which of the two rotating phasors 
they neglect. If one tries to transfer the flow of time into complex 
plane one ends up at Feynman's silly suggestion that there is a forward 
and a backward time, simultaneously.

I disagree with experts of set theory on a flaw being at least as old as 
Buridan's famous donkey. In order to benefit from absence of redundance, 
FCT has to be defined not as a particular case of FT within IR but 
independently (and as I would like to stress equal in status)  within 
*IR+. Mathematicians did not object when I said *IR+ contains IR. I feel 
in debt to Hermann Kremer who pointed me to original work by Cantor and 
Hilbert. I realized that Cantor's diagonal method was only consistent 
with infinity up to a n-th 'Grad'. Unfortunately, I have to confess that 
not even hyperreal numbers already offer equality between open and 
closed intervals of continuum.

In other words, I suggest attributing if necessary two different values 
or signs to the same point. While it is common but rather abitrary habit 
to except the point t=0 from sign function, I imagine past (-infty, 0) 
and future (0, inft) - not [0, infty) - to join each other without any 
neutral point 0 in between. Both past and future own zeros, each with 
the appropriate sign. The controversy might have serious consequences 
for mathematical tenets. Because my puzzle surprisingly comes completely 
out, I am convinced to be correct. If not, it would not at all impair 
FCT and natural spectrogram because at the sliding origin t=0, 
theoretically infinitely many new values emerge each moment in permanent 
succession.

Using Matlab, I managed to demonstrate that the natural spectrogram 
actually reaches best similarity with the natural solution, and its 
resolution does not just outperform the uncertainty principle as does 
audition too. Its resolution is in principle not subject to any limit.
In all, you might be sure that the natural spectrogram is well founded.

Eckard Blumschein

Hi Rune.

Sorry about the late reply. I've been away since thursday.


Rune> There are several meanings here. "I have implemented MUSIC as a
Rune> homework computer assignment in class" is one, "I have designed
Rune> a special variation of MUSIC that take advantage of some exotic
Rune> condition hardly ever found in practice" is another, "I tested
Rune> MUSIC with syntentic signals with different SNRs" is yet
Rune> another. The use I don't see is as in "I used MUSIC in an
Rune> unsupervised analysis system,applied to real-world measurements
Rune> in production applications".

Okay, that's what I thought you meant. Audio and speech modeling and
coding are exactly such applications and I know that at least ESPRIT
works very well in such a setting. Maybe it is also important to point
out that in these applications, people are usually happy if some
modeling or coding error is minimzed. Thus, the physical
interpretation of the parameters may not be all that important. I
guess that is a big difference from your applications and stuff like
DOA.  
 
Rune> Sure. That's why I haven't published anything. I find it not
Rune> worth the while to squeeze al relevant background, examples and
Rune> analysis etc into 10 or 12 pages, which usually is the budget of
Rune> an article.

Complex experiments are also often very hard to describe in a nice,
concise way. I consider it a big problem in audio coding, that the
experiments are typiccaly so complex, that there is no way, that you
can reproduce the results in a reasonable amount of time. Often people
leave out (or forget to mention) important details.

>> For audio coding applications (my field), for example, the
>> computational complexity of these algorithms compared to the
>> alternatives may be prohibitive. The difficulties in incorporating
>> perception may be another reason for not using them (although it
>> has been done for ESPRIT). Anyways, my point is just that there may
>> be many reasons for the lack of papers.
Rune> My point exactly. I found several problematic aspects of these
Rune> types of methods when I worked with them. The authors of journal
Rune> articles appear not to be aware of the problems.
 
I don't know if I really see that as a huge problem. If an estimator
can be shown to have nice properties lige low variance and low bias in
a synthetic setting, then I would say that we have good reasons to put
an effort into making it work in an application also. But that can, as
you point out, be difficult and usually takes a lot of expertice
within the area of the application. What can be frustraing, though, is
when the people working on the purely theoretical part of it don't get
the problems and just keep writing paper after paper after paper on
the subject.

Rune> That's because they are. Unfortunately, one have to be very
Rune> familiar with both linear algebra and vector spaces to work with
Rune> these types of methods. Otherwise, they fail. Big time. That
Rune> happens even with people who do know linear algebra and vector
Rune> spaces...

Yeah, sometimes I also get the feeling that people are happy about
these methods because they are cool from a mathematical point of
view. This also goes for subspace-based speech enhancement (denoising
or whatever you want to call it). The mathematics look way cooler than
the results sound ;)

>> MUSIC also seems to be fairly popular in estimation problems in
>> communication. At least I know people who use it in such
>> applications. Again, I think that computational complexity is the
>> reason that there haven't been many papers in the past on
>> applications in that area.
Rune> Believe me, numerical complexity is the least problem with
Rune> MUSIC.

I do believe you. But complexity may be a very good reason that people
in digital  communication haven't really looked into it until
recently. 
 
Rune> Perhaps not specific but such assumptions are qruical in getting
Rune> half-relevant results. Try, for instance, to implement MUSIC
Rune> with a signal covariance matrix of order 8. Then generate a
Rune> synthetic signal, SNR of your choosing, that comprises 10
Rune> complex exponentials at arbitrary frequencies. How does MUSIC
Rune> perform? What indication do you see that MUSIC has broken down?
Rune> What will the cost of such a break-down be in a real-world
Rune> applications?

Yeah. That is a big problem with MUSIC. Like I also said in a
discussion we had earlier on, the problem of chosing the right orders
of the covarinace matrix and the subspaces are not easy. At least in
my applications, you can never really know in advance how many
sinusoids there are going to be.

Generally, the art of making the right assumptions is the art of
engineering, at least in estimation. As always, it's a compromise. 

Rune> I would go a bit further. These algorithms are so cumbersome
Rune> that one should not expect them to work at all. Now, that
Rune> applies to a certain extent to class-room (or lab) excercises
Rune> with synthetic data, but the huge problems appear when you try
Rune> to take the methods out of the computer lab and try to use the
Rune> methods with real data.

That's a good point. At my university (Aalborg University in Denmark),
teaching is group and project-based which means that usually people do
end up working on real-life signals, so the students do get practical
experience working with signal processing algorithms, for example. The
downside is that we don't have as many lectures as other places do.

>> My guess is that the reason that you don't see many papers
>> concerning the problems in applications is, that tractable
>> experiments showing these kinds of problems are not very easily
>> made and evaluated in a formal way. Besides, who publishes negative
>> results (although negative results may prevent others from making
>> the same mistakes)? :-)
Rune> Exactly. So we agree one should expect some sort of bias in the
Rune> literature. It appears we agree on most points.

Indeed.

-- 
/Mads (http://kom.aau.dk/~mgc)

Randy Yates wrote:
> 
> A single sinusoid, well even I can estimate that.
> But simultaneously estimating multiple sinusoids of unknown frequencies???

Yes, this is not hard if you know the # of frequencies in advance, and
not *that* hard even if you don't but can bound it to a reasonably
small number (say, dozens).  It's just parametric time series modeling.
There's a wide literature on it, but I'm most familiar with the Bayesian
literature.  Bretthorst shows how to do it from this point of view in
his book, which is available online:

http://bayes.wustl.edu/

(See the G. Larry Bretthorst section near the bottom.)  If you email
him he will probably even have free software for you to do it.  I have
implemented it in Python (as part of a NASA project developing software
for analysis of astrophysical time series and other data) and will be
releasing that implementation this summer.  

In a nutshell---if the frequencies are well-separated, the periodogram
(the power spectrum, thought of as a continuous function of frequency)
is a suffucient statistic for estimating the frequencies (they are just
the frequencies of the peaks).  But for close frequencies, you have
to do something more sophisticated with the DFT than just look at its
magnitude; the refined procedure uses phase information and can reliably
measure frequencies, amplitudes, and phases for sinusoids much closer
than the Fourier frequencies, if only the S/N is high enough.

The approach is not limited to sinusoids; in fact, Bretthorst's main
practical application has been to analyzing NMR data (exponentially
damped sinusoids).  In that case a fast algorithm is still possible
by apodizing the data before the FFT.  I actually got into this for
developing methods for detecting extrasolar planets via doppler shifts
in spectral lines; in our case we
use a basis that isn't sinusoids, but instead the functions that
describe the line-of-sight velocity of a Keplerian orbit.  The data
are not evenly sampled; this combined with the Keplerian shape means
we can't use FFTs, but the data sets are small and the approach works
fine doing the sums the "slow" way.

Rune Allnor wrote:
> 
> "Have been used" is a vague term. I am interested in seeing these types
> of methods being used in a running environment where there are
> requirements to fail-safe processing and robust performance.

If you buy an NMR machine from Varian corporation (the kind used for
chemical analysis, not medical imaging), it runs Bretthorst's algorithm
for you.  They hired him to write the software.  It works incredibly well
in such a real-world environment, enough so that it was even covered as
part of a news item in *Science* a few years ago:

  Malakoff, D. 1999, Bayes offers a `New' Way to Make Sense
  of Numbers, Science, 286, 1460--1464
  http://www.sciencemag.org/cgi/content/full/286/5444/1460
  (I don't know if this URL has limited access...)

If, however, the frequencies or phases are changing in time in a
complicated way, you have to do more complicated things.  Actually,
Larry has worked on this (for modeling astrophysical sources) and has
some impressive algorithms, but they are a lot more complicated than the
"Bretthorst algorithm" I described above, which uses nothing other 
than the DFT.

I have a paper describing the Bretthorst algorithm implemented for
audio testing (with sums of sinusoids) if any readers would like to see
some examples of the method in that context.  Email me.

Cheers,
Tom Loredo

-- 

To respond by email, replace "somewhere" with "astro" in the
return address.

Tom Loredo <loredo@somewhere.cornell.edu> wrote in message news:<40900A32.8B6DB7F2@somewhere.cornell.edu>...
> Randy Yates wrote:
> > 
> > A single sinusoid, well even I can estimate that.
> > But simultaneously estimating multiple sinusoids of unknown frequencies???
> 
> Yes, this is not hard if you know the # of frequencies in advance, and
> not *that* hard even if you don't but can bound it to a reasonably
> small number (say, dozens).  It's just parametric time series modeling.
> There's a wide literature on it, but I'm most familiar with the Bayesian
> literature.  Bretthorst shows how to do it from this point of view in
> his book, which is available online:
> 
> http://bayes.wustl.edu/

Thanks for posting the link to the book. It looks interesting. 

Yesterday I gave a lecture in of a course we give here on statistical 
signal processing. My main objection against parametric frequency 
estimators (or any other parametric model, for that matter) is that 
some "clairvoyance" is required from the designer/user. 

As you say, knowing the number of frequencies in advance is of great 
help. How often does it happen, in real life, that you know that? 
In "my" seismic applications I have all the data available when I 
start the analysis. Because of that, I can do certain preliminary 
tests based on standard DFTs. Add some intuition and previous experience, 
and I can give a rough opinion about what waves and phenomena will be 
found in the data sets from a given area. However, I have discussed 
other types of applications of frequency estimators where one does 
not have this prior knowledge of what goes on, and where one does not 
have the opportunity to inspect and assess the data before embarking on 
an analysis using frequency estimators. My gambling records, minuscule 
as they may be, clearly prove that I am not a clairvoyant person. 
I can not tell beforehand how many narrow-band frequency lines, if any, 
will be present in the signal. For the most part, I can't even tell 
what model is best before testing models with known reference data. 
And I certainly don't know how to link up one particular parametric 
model with any given observation, unless doing several tests, 
evaluations and interpretations. And even then I may be plain wrong. 
 
To make a short story long, I'll describe a project I was involved in. 
Many years ago, an oil company did a marine seismic survey where shear 
waves were recorded. The shear waves that had bounced off the gas 
reservoir deep in the sea floor, were clearly visible in the recorded 
data. The problem was, where did those shear waves come from? The 
seismic source went off in the water column and could only generate 
pressure waves (P waves). The shear waves (S waves) had to be generated 
by P waves being converted somewhere. There were two competing 
hypotheses:

1) The S waves were generated when the P waves from the source hit the 
sea floor, then the S waves traveled down to the gas reservoir, bounced 
off, and the reflected S waves were recorded. This was known as the 
"PSS hypothesis".

2) The P wave from the source penetrated the sea bottom, traveled as 
P waves to the reservoir, got converted to S waves at the target, and 
the S waves propagated back up to the sea floor to be recorded. This 
was the "PPS hypothesis".

When dealing with sedimentary rocks, the rule of thumb is that the 
S wave velocity (Vs) is half the P wave velocity (Vp). Vp can be 
measured by studying certain types of waves in the seismic data. 
The next rule of thumb is that Vs in non-consolidated sediments is 
lower than in consolidated rocks. Given that Vp was measured to 
1800 m/s, we apply the first rule of thumb to find that Vs would be  
900 m/s in the top sediments, if they were to be regarded as 
consolidated rocks. Which we know they are not. The second rule of 
thumb is rather vague. How much lower should we choose the Vs to be 
for the non-consolidated sediments? Should we choose Vs=850 m/s? 
Vs=600 m/s? Vs=50 m/s? 

During the initial analyses, no one knew. Some expert chose to use 
600 m/s, based on these rules of thumb and some gut feeling. When 
they put these numbers into the equations, they found that the physical
conditions for generating S waves in the data supported hypothesis 1, 
that S waves were generated at the sea floor. But they couldn't quite 
get the numbers to add up for the rest of the analysis.

At that time, I was invloved in a student project at the university, 
on estimating Vs just below the sea floor. I got to test "my" method
with the seismic data, and came up with estimates for Vs in the range 
100 m/s - 300 m/s. Way lower than the initial "guesstimates" of 
600 m/s. All of a sudden, the PSS hypotheses had to be rejected, and 
one had to concentrate on the PPS hypotheses. With the new numbers, 
everything started coming together.

Note that the Expert who provided the initial estimates did everything 
right: He used what measurements were available to him, he used 
the rules of thumb that are generally acknowledged. He just missed 
on the scale. He guessed that Vs in sediments are half or two thirds
of Vs in consolidated rocks. With 20/20 hindsight, he should have 
guessed one fifth of the Vs of consolidated rocks. Had he done that, 
however, he would probably faced severe criticism. Guessing numbers 
an order of magnitude lower than the 900 m/s he started from, 
could probably not be defended without other means of justification.
Which were not available until my results were available one year 
later. 

After that initial test, I worked with the company for five years, 
developing and implementing high resolution analysis methods both 
for frequency estimation and parametric descriptions of the sea bed. 
I saw, and learned the hard way, that everything depends on getting 
those initial assumptions and models right. Achieving that requires
either meticulous preparations or some sort of clairvoyance, some 
times even both. If you miss at this very first stage, you very 
quickly find yourself in very big trouble.

So, in a very breaf summary,  I find it very hard to see how to use 
frequency estimators in situations where I can not inspect and test 
the data beforehand, and then decide whether it's worth the effort 
to apply a frequency estimator.

Rune

Hi Tom,

your post  and www pages are very useful. Thanks a lot. I am looking for
something like this.

Regards,
Michal Kvasnicka
"Tom Loredo" <loredo@somewhere.cornell.edu> p&#4294967295;se v diskusn&#4294967295;m pr&#4294967295;spevku
news:40900A32.8B6DB7F2@somewhere.cornell.edu...
> Randy Yates wrote:
> >
> > A single sinusoid, well even I can estimate that.
> > But simultaneously estimating multiple sinusoids of unknown
frequencies???
>
> Yes, this is not hard if you know the # of frequencies in advance, and
> not *that* hard even if you don't but can bound it to a reasonably
> small number (say, dozens).  It's just parametric time series modeling.
> There's a wide literature on it, but I'm most familiar with the Bayesian
> literature.  Bretthorst shows how to do it from this point of view in
> his book, which is available online:
>
> http://bayes.wustl.edu/
>
> (See the G. Larry Bretthorst section near the bottom.)  If you email
> him he will probably even have free software for you to do it.  I have
> implemented it in Python (as part of a NASA project developing software
> for analysis of astrophysical time series and other data) and will be
> releasing that implementation this summer.
>
> In a nutshell---if the frequencies are well-separated, the periodogram
> (the power spectrum, thought of as a continuous function of frequency)
> is a suffucient statistic for estimating the frequencies (they are just
> the frequencies of the peaks).  But for close frequencies, you have
> to do something more sophisticated with the DFT than just look at its
> magnitude; the refined procedure uses phase information and can reliably
> measure frequencies, amplitudes, and phases for sinusoids much closer
> than the Fourier frequencies, if only the S/N is high enough.
>
> The approach is not limited to sinusoids; in fact, Bretthorst's main
> practical application has been to analyzing NMR data (exponentially
> damped sinusoids).  In that case a fast algorithm is still possible
> by apodizing the data before the FFT.  I actually got into this for
> developing methods for detecting extrasolar planets via doppler shifts
> in spectral lines; in our case we
> use a basis that isn't sinusoids, but instead the functions that
> describe the line-of-sight velocity of a Keplerian orbit.  The data
> are not evenly sampled; this combined with the Keplerian shape means
> we can't use FFTs, but the data sets are small and the approach works
> fine doing the sums the "slow" way.
>
> Rune Allnor wrote:
> >
> > "Have been used" is a vague term. I am interested in seeing these types
> > of methods being used in a running environment where there are
> > requirements to fail-safe processing and robust performance.
>
> If you buy an NMR machine from Varian corporation (the kind used for
> chemical analysis, not medical imaging), it runs Bretthorst's algorithm
> for you.  They hired him to write the software.  It works incredibly well
> in such a real-world environment, enough so that it was even covered as
> part of a news item in *Science* a few years ago:
>
>   Malakoff, D. 1999, Bayes offers a `New' Way to Make Sense
>   of Numbers, Science, 286, 1460--1464
>   http://www.sciencemag.org/cgi/content/full/286/5444/1460
>   (I don't know if this URL has limited access...)
>
> If, however, the frequencies or phases are changing in time in a
> complicated way, you have to do more complicated things.  Actually,
> Larry has worked on this (for modeling astrophysical sources) and has
> some impressive algorithms, but they are a lot more complicated than the
> "Bretthorst algorithm" I described above, which uses nothing other
> than the DFT.
>
> I have a paper describing the Bretthorst algorithm implemented for
> audio testing (with sums of sinusoids) if any readers would like to see
> some examples of the method in that context.  Email me.
>
> Cheers,
> Tom Loredo
>
> -- 
>
> To respond by email, replace "somewhere" with "astro" in the
> return address.