1. The Ring Detector
2. How Particle Sizes and Measurement Angles are Selected
3. Origin of the Matrix Formulation – It’s just algebra
4. Inversion of Data – the Information Content in the Data |
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1. The Ring Detector
As mentioned in the foregoing article, laser diffraction relies on measuring the light scattering pattern originating from the suspended particles. The light scattering measurement is done by use of ring detectors. That is, each detector is a circular ring of some width. In the following, these will be referenced simply as rings. For mathematical reasons, the rings have inner and outer radii that increase in a fixed ratio. Thus, ring radii increase in a geometric progression. It follows that logarithm of the ring radii increase linearly. For this reason, the ring radii are often referred to as log-spaced. Because the inner-most rings are very small and very thin – the inner ring has a 100 micron inner radius and is 18 micron wide – their surface area is small. In contrast, the outer-most rings are much larger and very wide – the outer ring has an inner radius of 16,950 microns, and a 20,000 micron outer radius, which makes it a very large area. |
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| This change in areas of rings is arranged to reduce the dynamic range of outputs of the 32 rings: Light intensity as a function of angle varies over several orders of magnitude (plot a below) and the log-spacing of rings reduces the photo-current out of the ring array to about 3 orders of magnitude (plot b below). These photocurrents are amplified, passed through an A-D converter and recorded as raw data on the LISST data logger. They become the 32 bits of primary data, which are solved like algebraic equations to construct the PSD. This is the inversion step. (These photocurrents also represent the small-angle forward light scattering property volume scattering function (VSF). LISST instruments remain the only field devices that can measure the VSF at very small angles, which are important for underwater imaging.) |
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| a) shows the actual scattering from 3 differently sized spherical particles. Scattering intensity is on the ordinate axis. It is readily seen that the scattering intensity varies 3-4 orders of magnitude. b) shows the same scattering as seen by the 32 LISST rings. |
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| Why is this method called laser diffraction? That is because of a convenient result: at small forward angles, diffraction of light dominates the measured scattering signal. This makes the composition of the particles, i.e. their refractive index, unimportant. As a result, the method is usable for PSD determination of all types of particles, no matter what they are made of – from pharmaceuticals, cements, chocolates, coffee, sediments, abrasives, to water droplets in the sky. |
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| To invert measured multi-angle scattering, one is essentially finding a PSD that has the same or nearly the same multi-angle scattering as the measurement. This means that we must know the multi-angle scattering from a unit quantity (say volume, or mass, area, or number) of particles of any size. In other words, we need to have a model for light scattering. Because diffraction dominates small angle scattering, the model that was chosen initially in then 1970’s when laser diffraction was developed, was light diffraction through an aperture. |
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Initially, the light scattering model was intended for spheres. It was known from Lorentz-Mie theory – developed in 1908 – that at small angles diffraction dominates the total light scattering signal. And diffraction of a particle could be replaced by diffraction through a round hole, an aperture. The function describing this process is a very famous and widely known function in optics. It is called the Airy function. To persons familiar with acoustics, this is the same as the shape of far-field acoustic beam formed from a piston transducer. It has a shape involving the zero order Bessel function, J0, the wavenumber of light  , and the particle radius a. The Airy function at an angle,  , is: |
 (1) |
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| This is a very convenient form. One only needs to calculate this function once for a range of values of the argument x = ka. Thus for any particle size a, the scattering at an angle can be calculated or looked up from a single table. For this convenience, this was the model used for scattering from spheres. Thus constructing a PSD from multi-angle scattering data I() amounted to finding concentrations of particles of a range of sizes from some low size alow to a high size ahigh that would fit the data. Because there are 32 data points in a LISST measurement 32 such sizes can be found. Thus, one could find 32 sizes in some range. The next question is what sizes should be the lowest and highest. |
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2. How Particle Sizes and Measurement Angles are Selected
Note that the Airy function depends on one single parameter, x = ka. [the Airy function looks a lot like [sin(x)/x]2]. It has very well defined maxima and minima. In order to observe a particular size, it makes sense to capture the maximum of its Airy function, which occurs at x = ka = 1.372. This would imply that the maximum particle size would be related to the minimum angle, and vice versa. In other words, the sizes aj were chosen such that for the 32 angles i [where i and j could take values 1 to 32]: |
| k aj 33-i = constant [i,j take values from 1 to 32] (2) |
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| The value of this constant is set to 2 by Sequoia in its instruments. But, it is a free parameter to an extent. The consequence, in any case, is that the maximum size is defined by 2/[k1], i.e. the smallest angle, and the smallest size by 2/[k32]. |
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3. Origin of the Matrix Formulation – It’s just algebra
The scattering measured at angle i can be written as the sum of contributions from concentration Cj of particles of size j. In mathematical terms: |
 (3) |
| What this equation says is that the scattering at any angle (i.e. the light intensity on any ring detector) is the sum of scattering from particles of all sizes, weighted by their concentration C. This is important because IT IS OFTEN MISTAKENLY CONCEPTUALIZED THAT ALL LIGHT ON RING i IS RELATED TO CONCENTRATIONS OF PARTICLE OF SIZE 33-i. NOT CORRECT. ALL PARTICLES SCATTER LIGHT AT ALL ANGLES. THUS ANY RING SEES LIGHT FROM ALL PARTICLES. THIS IS WHY INVERSION IS NECESSARY TO CONSTRUCT THE PSD.
First, we rewrite Eq. (3) as: |
 (4) |
| which simply states what we said in words, “light on detector i is the sum from all particles of sizes aj weighted by their concentration Cj.” This is the matrix product written as follows: |
 (4a) |
where now we use  and  to indicate the fact the light scattering intensity and concentration are both 32-element vectors. K is a 32 x 32 matrix, which has elements  , the Kernel Matrix. The 2 knowns in this equation is the light scattering intensity vector and the Kernel Matrix K. By inversion, we find the concentrations of different sizes in a mix of particles that would generate the same scattering as was observed. |
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We have so far not defined the units of concentration . As written in Eq.4, is number of particles. However, if we replace the Airy function by Airy function divided by volume of the size class of particles, then can be replaced by volume concentration of particles  . This would mean rewriting Eq.4 in the form: |
 (4b) |
| and as a matrix product: |
 (4c) |
| If the units are concentration in area of particles per volume of water (area concentration), then the Airy function would be normalized by a2 instead of a3 in Eq.4b. In short, the kernel matrix is computed based on the units of concentration. |
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| To recap, if we wish to cover size range alow to ahigh, we choose the angles accordingly, measure the scattering and invert it to get the size distribution. No further information is needed about particles – e.g. what they were made of, and even if they were round or not (which was a leap of faith anyways). The resulting inversion creates a PSD of equivalent spheres for the measured multi-angle scattering. Actually, it was not equivalent spheres, but equivalent apertures. Ignoring this detail permitted use of this method, by now called laser diffraction, to measure particles in industry and academia. |
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4. Inversion of Data – the Information Content in the Data
This part of the development is highly mathematical. We summarise the difficulties and the results. First, if one simply performs the inverse as |
 (5) |
| bad things happen: One can see negative values in the concentration; the results can have spikes or zeros, all unpleasant and not physically possible. The reason for this is two-fold: data are never perfect, and the kernel matrix K is “ill-conditioned”, i.e. has a high condition number. (The condition number is the ratio of the largest to the smallest Eigen values of the kernel matrix). To circumvent this, the mathematical field of “inverse theory” has arisen, with a strong motivation initially from atmospheric sciences. |
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| Over the years, numerous inverse methods have been developed that can be generally applied. For example, some methods searches for non-negative solutions, which in some least-squares way, match the data best, or perhaps they impose a condition of smoothness on the result and so on. The very first LISST instruments were shipped with processing software that was based on such a smoothing inversion algorithm, the Philips-Twomey algorithm. |
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| Smoothing, of course, is contrary to RESOLUTION. Thus, in the late 1990’s several papers appeared in journals pointing out the poor resolution of LISST PSD’s. Subsequently, Sequoia developed a solution which preserves high resolution and remains non-negative using a non-linear iterative algorithm – the NLIA algorithm. All processing software since 1999 (SOP version 4.x and above) contains the NLIA solution. This is Sequoia’s only ‘trade secret’. |
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| Consequently, since 1999 Sequoia’s NLIA inversion can resolve particles two size classes apart (i.e. diameters differing by factor 1.44). The figure below serves to illustrate the difference between the original smoothing Philips-Twomey algorithm (1995-1999) and the high-resolution NLIA algorithm (1999-present). The scattering signature for an artificial size distribution with particles in size bins 10, 20 and 28 only was inverted using the Philips-Twomey and the NLIA algorithms, respectively. It is evident that the NLIA inversion algorithm perform much better than the Philips-Twomey. |
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| In short, the resolution of the LISST data now is not as low as reported in an often cited paper by Traykovski et al. (1999): A laboratory evaluation of the laser in situ scattering and transmissometery instrument using natural sediments. Marine Geology 159:355–367. This paper should only be cited with respect to the PSD resolution if you are inverting LISST data using LISST-SOP version 3.x or below, which used the Philips-Twomey solution. All versions of the LISST-SOP from 4.x and above uses the NLIA solution. |
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| Questions to this article? Email us! |
| Updated 4/18/2011 |