The scattering properties of aquatic suspended particles have many optical
applications. Several data inversion methods have been proposed to estimate
important features of particles, such as their size distribution or their
refractive index. Most of the proposed methods are based on the Lorenz–Mie
theory to solve Maxwell's equations, where particles are considered
homogeneous spheres. A generalization that allows consideration of more complex-shaped
particles is the

Light particle interactions, usually wavelength dependent, cause observable
optical phenomena (e.g., changes on the ocean color or light extinction with
depth) that allow assessing the composition of small particles (e.g.,
phytoplankton, sediment, or microplastics) in the water column

Maxwell's equations are the basis of theoretical and computational methods
describing light interaction with particles. However, exact solutions to
Maxwell's equations are only known for selected geometries. Scattering from
any homogeneous spherical particle of arbitrary size is explained
analytically by the Lorenz–Mie (also known as Mie) theory

Procedure used to analyze the accuracy of the inverse models.

Both the Lorenz–Mie and

Several inverse models to retrieve the refractive index from optical
measurements can be found in the literature. For instance, a single equation
based on the Lorenz–Mie theory was used by

In this paper, the above refractive index retrieval models are reviewed and
tested against simulated data in order to analyze their accuracy when
modeling real (and usually complex-shaped) particles suspended in water, such
as phytoplankton. The comparison has been done following the three steps
presented in Fig.

The simulated examples are implemented using complex refractive indices and PSDs similar to those found in nature for phytoplankton species. Since phytoplankton particles exhibit a wide variety of shapes, each example has been provided with a different outline accounting for a homogeneous sphere, a coated sphere, and a homogeneous cylinder. None of these idealized shapes is an exact representation of real algae presenting cell walls, chloroplasts, vacuoles, nuclei, and other internal organelles, each one with its own optical properties. However, they can be considered as a first approximation suitable for the purposes of the tests presented in this contribution.

It must also be noted that the models are fundamentally different. The model
developed by

In order to establish the foundations of the work here presented, Sect. 2
reviews the formulation used to obtain the IOPs from the Lorenz–Mie and

Algal assemblages are typically polydispersed with regard to size, and can
be described by a PSD

Lorenz–Mie and

Scattering can be further characterized in terms of the angular distribution
of the scattered light using the volume scattering function (

This normalization implies that the backscatter fraction can be computed using the volume scattering phase function as

The

In this section, a review of the different approximations to retrieve the
refractive index (inverse models) is presented. Each model is named after the
lead author of the publication. The complex refractive index

The Twardowski model

The link between

This model is based on the methods presented by

The central value

The exact value of

This methodology was subsequently simplified by

The model presented by

After the evaluation, the algorithm stops if a satisfactory fitness level
or a maximum number of generations is reached (each generation is a new
vector of solutions). If the convergence condition is not fulfilled, the best
solutions are selected and taken apart. Part of this elite is then recombined
(crossover) and randomly mutated to provide genetic diversity and broaden the
search space (crossover and mutation introduce the diversity needed to ensure
that the entire sample space is reachable and suboptimal solutions are
avoided;

Since the Lorenz–Mie and

Flow chart for the genetic algorithm.

The main advantage of this method is that it can be easily adapted to
different Lorenz–Mie or

This section has been summarized in Table

Summary of the refractive index retrieval models.

The models described in the previous section are used here to retrieve the refractive index of well-known particles in order to determine their accuracy by means of the averaged relative error defined as

First, Sect.

A concentration of 100 spherical particles per cubic millimeter, with the PSD shown in
Fig.

The complex refractive index in Fig.

The spherical-shaped particle idealization was first examined with the
Twardowski model. The use of Eq. (

This model overestimates both the real and imaginary parts for all analyzed
spectra (Fig.

Postulated and estimated refractive indices using

In order to implement the genetic algorithm described in Sect.

One disadvantage of the genetic algorithms is that they are relatively slow
and require more computation time than other optimization algorithms, as they
compute the fitness function many more times. Other optimization algorithms
were also applied to determine whether similar results can be obtained with a
significant reduction of the computation time. However, since none of them
led to any meaningful improvement, no further description is provided. As a
single example, Fig.

In order to use the IOPs of a two-layered spherical particle that emulates
actual phytoplankton organisms, its complex refractive index was generated
using the description presented in

Postulated

PSD simulating an isolated culture.

The above set of IOPs can now be used to estimate the corresponding complex
refractive indices. First, the genetic algorithm is used in order to see
whether a basic shape, such as a homogeneous sphere, is useful when modeling
more complex particles. If coated particle models do better at characterizing
the optical properties of general phytoplankton species, as stated in

The genetic algorithm model, previously employed to retrieve the refractive
index of spherical-shaped homogeneous particles, was used for the
spherical-shaped coated particles. The same configuration was used, i.e., an
initial vector of 2000 solutions over 10 generations and 50 and 20 % of
probabilities for crossovers and mutations, respectively. The real part of
the homogeneous case lies between the real values for the inner and outer
regions in the coated case (Fig.

Volume scattering function using the estimated refractive index for spherical-shaped coated particles.

The Bernard model, described in Sect.

Postulated and estimated

One possible modeling improvement would be to couple the genetic algorithm, which showed a reasonable performance when applied to homogeneous spherical particles, with the BART code (instead of the BHMIE code) in order to estimate the two complex refractive indices. However, the results would hardly be constrained as the solution has more degrees of freedom (the two refractive indices with their real and imaginary parts give 4 degrees of freedom) than the available data (2 degrees of freedom, corresponding to the attenuation and scattering coefficients), i.e., this is an unconstrained (ill-posed) problem. Alternatively, the Bernard model could be combined with the genetic algorithm to increase the probability of convergence. In this case, the inner refractive index is again estimated using the Bernard model and the outer refractive index is then obtained with the genetic algorithm (coupled to the BART code). In this case, the genetic algorithm only has to find a solution with 2 degrees of freedom (the real and imaginary parts of the outer refractive index).

This method was applied to the coated particle example using the coefficients
of Fig.

Postulated and estimated

As a final example, a cylindrical-shaped particle has been chosen. As
commented above, phytoplankton species usually present complex shapes, far
from perfect homogeneous or coated spheres. This is the case, for example, of
the centric diatom with cylindrical shapes (to name a few genera:

PSD for cylindrical-shaped particles.

As previously discussed, the simulated cylindrical particles are not exact
duplicates of the actual phytoplankton organisms, so it is useful to compare
the results using this and the coated sphere design (usually used on all kind
of phytoplankton shapes). As in previous examples, the postulated value of

Inner and outer

The genetic algorithm can be combined with the

The above methodology was applied using the same PSD as in Fig.

Volume scattering function obtained using the Bernard model combined with the genetic algorithm for cylindrical-shaped particles.

The average relative errors of the real and imaginary parts of the estimated
refractive indices, together with their respective volume scattering
functions, are shown for each method applied to the three case examples
considered in the previous section (Table

Averaged relative errors for each method.

Spectral backscattering for the three test cases: homogeneous sphere, coated sphere, and homogeneous cylinder.

In the homogeneous sphere example, the Twardowski model presents the highest errors, especially when comparing the volume scattering function. Although the Stramski model leads to complex refractive index errors considerably higher than the genetic algorithm (particularly for the imaginary part), comparable estimates of the volume scattering function are recovered in both cases. This implies that, for this particular example, there is no need of accurate refractive index estimates in order to obtain a suitable characterization of the scattering properties. However, the genetic algorithm performs with excellent accuracy for the refractive index retrieval.

In the coated sphere example, the genetic algorithm approximates the coated
particle to a homogeneous one with a single complex refractive index.
Therefore, errors for the inner and outer refractive indices cannot be
obtained; additionally, this method differs substantially when computing the
volume scattering function. Hence, if the optical response of coated spheres
was similar to the response of actual phytoplankton particles

Finally, the optical properties of homogeneous cylinders are not accurately
reproduced using coated spheres when their refractive indices are obtained
combining the Bernard model and the genetic algorithm. It is likely that the
optimal retrieval method would be the genetic algorithm using cylindrical
shapes to obtain accurate estimates of the refractive indices. However, this
involves using the slow

This study has pointed at three important lines for future research:

All the test cases are synthetic examples that, presumably, are simpler than real life. Further work is necessary in order
to study the performance of algorithms when using the optical properties of actual phytoplankton species and bulk oceanic measurements.
A more complex inversion method remains to be developed in order to deal with a mixture of shapes and refractive indices. This scenario
is currently dealt with using only spherical particles of different sizes (

When dealing with actual phytoplankton, a critical issue is the instrumental accuracy. The attenuation and scattering coefficients are
key inputs for all retrieval methods, in order to retrieve valid refractive indices. However, as stated by

The accuracy of the inversion methods could be improved by applying the

Ocean optics goes from research at a microscopic scale (as shown in this paper) to remote sensing, measuring the reflected or
backscattered radiation in large areas. The inversion methods based on Lorenz–Mie and

To conclude, the results presented in this paper and summarized in Table

The accuracy of different inverse methods retrieving refractive indices from the optical properties of small scatterers, and their particle size distribution, has been analyzed. To this end, three different synthetic examples were constructed, each one with a different shape and distribution. The selected shapes were homogeneous spheres, coated spheres, and homogeneous cylinders. The results indicate that the most accurate methods are those using a genetic algorithm to optimize the inversion, although they were also the slowest ones. In particular, an excellent agreement was obtained between the estimated and actual refractive indices and volume scattering functions for the homogeneous and coated sphere cases, and a fair agreement for the homogeneous cylinders. The results suggest that even better characterizations could be obtained for the actual phytoplankton optical properties. A next step should be the analysis of the performance of these methods when applied to measurements of isolated cultures of phytoplankton.

This work was supported by the Spanish National Research Council (CSIC) under the EU Citclops Project (FP7-ENV-308469), the MESTRAL project (CTM2011-30489-C02-01), and the CSIC ADOICCO project (Ref 201530E063). The authors would also like to show their gratitude to Laura Pelegrí, Jimena Uribe, Josep Lluís Pelegrí, and Miquel Ribó for the English revision, as well as to Emmanuel Boss and two anonymous reviewers for their comments and suggestions, which helped to enhance the quality of the manuscript. Edited by: E. Boss Reviewed by: two anonymous referees