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 MatlabĀ®. 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 This routine has been superceded by a more sophisticated function which can do just about everything.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.