Selection of Informative Wavelengths of Laser Sensing
It is obvious that the number of sensing wavelengths n in the given spectral range Л depends on how close is the separation between the chosen neighboring wavelengths, for example, and A/+1, at the given accuracy of the measurement instrumentation. Denote the smallest relative error of measurements in the spectral range Л as e and consider the problem on the acceptable closeness of two wavelengths at the sensing of atmospheric aerosol. Assume that the values of the optical characteristic ДА) are measured experimentally in the range Л. If Aft = /3(Ат) — в(А) is the increment of the optical characteristic в in the vicinity of the point A, then the smallest separation between At and A,+1 should satisfy the obvious inequality A в > ев (A). In References 34, 64, based on the analysis of polydisperse integrals, the validity of the following inequality was demonstrated:
where the minimum is taken for the pair of neighboring A and A +1 at all possible divisions of the sensing range Л. Here, A is a constant depending only on properties of the aerosol distribution (0 < A < 1). Thus, if A is some sensing wavelength, then the next wavelength should be chosen at a distance exceeding the value AA > eAA. Then, it is easy to estimate the acceptable number of wavelengths totally for the range Л. Denote the left and right boundaries of Л as Amin and Amax, respectively. Then the following relations can be used:
of which the former determines the sequence of the values A, while the latter is the maximum allowable number of readouts. These equations show that, regardless of the properties of microstructure, with an increase of A the readouts should be more and more rare, and the value of the corresponding interval AA is directly proportional to A. To use these equations, it is necessary to know A. Formally, A is the functional of the distribution s(r) and is determined by the equation
which can be used for numerical estimates. In the case of the particle size distribution in the form of the gamma function  (see also Chapter 1):
we can show the validity of the approximate equality A и 1/(a + 3), where a is the distribution parameter. It should be noted that the equations resented are approximate, but they are sufficient in the case when the assumption of the unimodal form of the sought distribution s(r) is allowable, for example, for the droplet spectrum in low-level stratified clouds. In the absence of a priori information on the character of s(r), the more rigorous estimation of the number of independent measurements can be performed following References 70, 79 within the framework of the Fourier analysis of continuous measurements.