Multi-frequency?

The sensors used to assess
zooplankton populations that I describe on this site employ
multiple frequencies. Why? Well, ONE of the ways to
estimate sizes and abundances of zooplankton is by
measuring backscattering at several frequencies. The
concept is simple, more or less. At any given frequency,
the intensity of the sound scattered by a volume of water
is proportional to the number of scatterers of each size
class multiplied by the expected backscattering intensity
of each size class at that frequency. Do this over numerous
frequencies and you can construct a set of equations in F
frequencies and M sizes. In matrix notation,

RN = X

where R is the matrix (F by M)
of expected backscattering intensities at the F frequencies
and the M sizes. X is the vector (F x 1) of measured
backscattered intensities (corrected for sample volume and
range and such), and N is the (1 x M) vector of unknown
abundances. I wrote a paper on this problem a long time ago
(see Greenlaw and Johnson 1983 on the downloads page). This
paper discusses the need for accurate models of
backscattering (matrix R), measurement issues, and solution
issues. There is some new work that I'll report on here
when I get a chance.

Is this the only way to estimate size-abundance of
scatterers? Of course not. I did some work on using
backscattering at different angles to do the same thing.
This is analogous to what optical guys do with small
particles -- they measure optical backscattering at a
couple of angles to estimate the mean size of the
particles. Works in acoustics as well. And there is some
indication that you could measure physical properties such
as density and compressibility at the same time.

I put this page up initially to make the 1983 paper
available. I will flesh this topic out as time permits,
however. After all, this is what I mostly did for my
professional career. You'd think I might have a thing or
two to say about it ...

(much later)

Users of our TAPS-6 gadget received a CD with analysis
programs written in Matlab®. These were written in the
early 1990's so the Matlab version is rather old. However,
the code should still work, even on modern versions of the
language. In order to make the CD complete and 'run-able'
on newer computers, I put a copy of the inversion program
we used in the UTILITIES folder. This function is called
NNLS, which is computer shorthand for Non-Negative Least
Squares and was available with the older versions of
Matlab. This routine has been superceded by a more
sophisticated function which can do just about everything.
Van Holiday started using this version when it came out and
he wrote some programs using it. I will pull them off his
computer and add them to this section one of these days.

Some history...I was in grad school at Oregon State
University in the 1970's. My major professor, Richard
Johnson, was working with a biologist, William Pearcy, on
acoustical estimation of the abundance and sizes of
mesopelagic fishes. My role was to help setup and to
calibrate the acoustic system, Dick did the data
processing, Bill obtained biological samples for
comparison. From a seminar at Asilomar in 1975, the notion
of using backscattering measurements at several frequencies
to estimate size-abundance of scatterers had arisen
(although the idea was previously suggested by Donald
McNaught in 1968 and 1969 in an obscure journal) A paper on
the mathematics of least-squares estimation was presented
at the seminar by Van Holiday. This method clearly required
a mathematical representation of the backscattering
properties of the scatterers and a paper on scattering
models was presented by Dick. Back at OSU, Dick began
diligently trying to use least-squares inversion to
estimate the swimbladder size distributions from our
multi-frequency acoustic data.

The problem was that, almost always, the solution vectors
contained negative abundances at some sizes. This was
clearly contrary to physical reality. Dick spent a
considerable time down in the library and at the bookstore
and eventually surfaced with a copy of __Solving Least
Squares Problems__ by Lawson and Hanson. In this book
they described an inverse procedure that let you constrain
the candidate solutions to non-negative vectors of
abundance. The program they presented (in Fortran, of
course!) was called NNLS. Dick obtained a copy of this
program from the authors and began using it to process our
mid-water backscattering data on his PDP-11. The results
were very encouraging. Van Holiday had been using a
least-squares program to process his data (explosive source
broadband scattering from fishes) but soon switched over to
NNLS as well.

Later on, when Dave Doan built the 21-frequency MAPS system
for Van, he used NNLS exclusively to process the
backscattering data from zooplankton. Van was working with
Rick Pieper (marine biologist at USC) during this time and
they accumulated a mass of both acoustic and biological
data. A grad student of Rick's, Jack Costello, later (1989)
did a comparison of these data sets to show that the
acoustic estimations were very good.

(An amusing side-note: I was working at Tracor in Austin at
the time and had a little bit of $ to process some
zooplankton acoustic data I had taken while still at OSU.
One set of data -- extremely sketchy -- was some HF
backscattering estimates from zooplankton in a freshwater
lake. I was trying to fit these data to a fluid sphere
model, with absolutely no success. The data simply didn't
oscillate nearly as rapidly as the model did and seemed to
keep increasing with frequency (like I saw with the
preserved copepods in my 1976 JASA paper). I had been
working at ARL/UT when Don Shirley was doing some
incredibly careful measurements on backscattering from
hollow aluminum spheres in water. These objects,
precisely-machined spheres, exhibited rapid and extreme
fluctuations in Target Strength with frequency due entirely
to the geometry of the targets. My zooplankters,
fresh-water copepods, were more blob-like than spherical,
so surely it was asking a lot to believe that the
circumferential waves which contributed so strongly to
backscattering in perfect spheres would exist on a
non-spherical blob with legs. Thinking about the fluid
sphere solution I was using (Anderson 1950), I realized
that the first two terms were quite likely to exist no
matter the actual shape of the scatterer. The 'zeroth' term
reflects the compressibility of the object as the incident
pressure wave compresses it and it rebounds. The 'first'
term reflects the density of the object as it moves back
and forth in response to the incident wave. Maybe keeping
just these terms and ignoring all higher terms was
physically justified. Turns out, this model fit my data
rather well. I faxed a figure to Dick at OSU to show him
what I'd found. We talked about it on the phone and I
convinced him that this wasn't an extreme stretch to
believe.

A day or two later, I got an excited call from Van in San
Diego. "Guess what?" he said, "I've got a great new model
that works really well with copepods!")

I have put the code for the TAPS-6 system on this website.
Look under TAPS-6 and Inverse Code. I would like any
feedback you might have on these programs, especially other
uses of the code.