# Iterative Method for Solution of the System of Lidar Equations

Let us dwell briefly on issues associated with construction of an algorithm for numerical solution of system of Equation 5.35. For this purpose, we use the formal operator of mutual transition introduced in References 34, 64 as *W = K _{ext}K_{n}*

^{1}or

*W = K*In this case, system (5.34) can be rewritten as

_{n}K-_{x}].

Then we transform (5.36) into the discrete form, assuming that digitization of the lidar signal at every wavelength € Л yields a set of equidistant readouts *P _{ki} = P(z_{k},* X,) with a step Az

*= z*

_{k}+_{1}*—*z

_{k},

*(k =*1, 2, ...

*Nz).*Then, for every

*k*we have a system of lidar equations

where *S _{ki} = P_{ki}z_{k}IP_{oi}G_{i}* is the square-amplified backscattering signal;

*Uj*are quadrature coefficients.

If system (5.37) is solved consecutively starting from *k =* 1, then for the point *z _{k }*all the previous values of optical characteristics up to

*(3^*and

_{k—1},i*в*are known. In this case, it is more convenient to represent Equation 5.37 in the form [64,68]:

_{е1а>к—}и

where

*r _{ki}= Az(ft_{ext}?__{l},_{i} + @_{ext},k,i)l*

*2*is the optical thickness of the k-th layer of the sensed cloud.

To solve system (5.37) along with vector Equation 5.36 for vectors |3_{n} and |3_{ext}, an iterative algorithm was proposed and mathematically justified in Reference 68. The scheme of this algorithm can be represented in the following analytical form:

where *m* is the iteration number; the operator *W*is the matrix analog of the operator *K _{e}xt K_{x}*

^{1}.

For sensing of optically dense scattering media, in particular, stratified low-level clouds, the increased resistance of the algorithms to the multiple-scattering noise is needed. Based on these requirements, the parametric modification of iterative scheme (5.39) was proposed in Reference 20. It was used in further calculations of the vectors |3_{n}(A;) and *в _{еХ}(()* for the chosen wavelengths of laser sensing

*.*