Spectral interpolation

Multi-channel measurements

Measurements of optical extinction coefficients are performed at a limited number of discrete wavelengths.

For certain applications, one needs to know also the extinction at wavelengths where no measurements were made. An example is the retrieval of atmospheric trace constituent concentrations from total optical extinction coefficient, measured at a particular wavelength. To find the contribution from the constituent alone, one must subtract the aerosol component at that wavelength.

Another application is the retrieval of derived aerosol parameters, which requires the knowledge of ß over a broad spectral range (from UV to far IR). An example is the aerosol particle number density, the particle size distribution mode radius and dispersion. Spectral interpolation of extinction coefficients and this radial inversion are closely related. On one hand, experimental data requires interpolation or fitting before spectral inversion can be carried out. On the other hand, the spectral interpolation problem becomes trivial, once the aerosol size distribution parameter are known.

Mie theory

In case of one spherical particle, the extinction cross section Q_ext is given by the exact Mie Theory. But in realistic situations, one has to deal with extinction by a large number of particles, with different sizes. Such a particle population can be characterized by a size distribution function f(r). In that case the extinction coefficient for a given wavelength is:

β(λ) = ∫_[0,∞[ Q(λ,r) f(r) dr (1)

where f(r) is the particle size distribution function. As experimental extinction coefficients are known at discrete wavelength we can write the equation above as follows:

&beta(λ_i) = ∑_j Q(λ_i,r_j) f(r_j) (2)

i=1 to M, where M is the number of measurement wavelengths;
j=1 to N where N is the number of particle size bins.

The expression above is a set of M equations with N unknowns. In this system of equations, extinction coefficients are known (measured) and extinction cross sections can be calculated from Mie Theory. Solving that system of equations is the radial inversion and is an ill posed problem. It is possible to overcome this numerical analysis problem. To simplify let us write the system above as following for experimental wavelength:

β_exp = Q_exp f (3)

if λ _min is the smallest experimental wavelength and λ _max the largest one, then it is possible to calculate interpolated extinction coefficients for wavelengths lying in the interval [λ _min, λ _max]. For wavelengths outside the interval mentioned above we have an extrapolation problem that we will not consider here. For wavelengths where we want to interpolate we can write the same equation as for the experimental wavelengths:

β_int = Q_int f (4)

By choosing the same vector f for both experimental and interpolated wavelength it is possible to get the interpolated extinction as a function of the experimental ones. By multiplying the two sides of (3) by the Q^T_exp we obtain:

Q^T_exp β_exp = Q^T_exp Q_exp f (5)

The particle size distribution function f can then be obtained my multiplying the two sides of the equation(5) by [Q^T_expQ_exp]^-1

[Q^T_exp Q_exp]^-1Q^T_exp β_exp = [Q^T_exp Q_exp]^-1Q^T_exp Q_exp f
= f (6)

Finally by replacing f in the interpolated system of equations (4) by its value obtained in (6) we get the system of equations that performs the spectral interpolation of extinction coefficients

β_int = Q_int [Q^T_exp Q_exp]^-1 Q^T_exp β_exp (7)

The system of equations above is very close to singular and can not be solved by Gauss elimination or LU decomposition. To deal with such problems we have to use singular value decomposition (SVD) techniques. SVD techniques are based on the following theorem of linear algebra [William H. Press et al 1992]: any MxN matrix A whose number of rows is greater or equal to its number of columns N, can be written as the product of an MxN column-orthogonal matrix U, an NxN diagonal matrix W with positive or zero elements and the transpose of an NxN orthogonal matrix V. The inverse of A is then given by

A^-1 = V . [diag(1/W_j)] . U^T (8)

So if one has to solve a system of equations

A x = B (9)

then the solution is given by

x = V . [diag(1/W_j)] . U^T . B (10)

Applying this result to the system (3) didn't lead in our case to a satisfying solution. Indeed the solution is not unique and we clearly have to incorporate further information about the desired solution in order to stabilize the problem and to single out a useful stable solution. In fact we have to apply regularization methods to the system of equations above which is clearly a discrete ill-posed problem. Although many types of additional information about the solution are possible, in principle, the dominating approach to regularization of discrete ill-posed problems is to require that the square of the norm or an appropriate seminorm of the solution to be small.

In our case we minimise

L = |Q_exp.f(r) - β_exp|² + ρ|f(r)|² (11)

Where ρ is the regularization parameter. It is possible to show [William H. Press et al 1992] that f(r) is given by

f(r) = (Q^T_exp.Q_exp + ρI)^-1 . Q^T_exp.β_exp (12)

Where I is the identity matrix. We obtain the interpolated extinction coefficients from:

β_int = Q_int [Q^T_exp Q_exp + ρI]^-1 Q^T_exp β_exp (13)

The choice of the regularization parameter is not easy and the direct inversion of matrix is in general not the best way to solve our problem. Therefore we used a regularization tool [Hansen,1994 ] to perform matrix inversion and to find the best regularization parameter. To summarize, each interpolation of a set of measured extinction coefficients includes the following steps:

compute Q_exp using experimental wavelenghts and selected particle radii r_i;
compute Q_int using wavelengths at which we want to interpolate extinctions and the same set of particle radii r_i as in the previous step;
perform matrix inversion SVD wise;
compute the best regularisation parameter ρ ;
and finally compute the interpolated extinction coefficients

In addition to the interpolated and experimental extinction extinctions coefficients, we also present coefficients that were fitted according to the Angström Law

β(λ) = α &lambda^ξ (14)

References:

William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling; "Numerical Recipes in Fortran: The Art of Scientific Computing"; Cambridge University Press (1992), pp 795-803.
Per Christian Hansen; "Regularization Tools: A Matlab Package for Analysis and Solution of Discrete Ill-Posed Problems"; Numerical Algorithm 6 (1994), pp1-35.