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Linear Approximation in Frequency Domain

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.

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Analytical solution of the two-frequency lidar inversion technique.

¹Meteorologisches Institut, Universität München, Theresienstrasse 37, D-80333 Munich, Germany. uh234aj@mail.lrz-muenchen.de

|March 8, 2008

Summary

This study demonstrates a unique solution for two-frequency lidar inversion, overcoming previous limitations. It provides an analytical method to determine aerosol extinction profiles and boundary values.

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Area of Science:

Atmospheric optics
Remote sensing

Background:

Two-frequency lidar inversion methods rely on assumptions about extinction coefficients and ratios.
Previous research questioned the uniqueness of solutions for boundary values and aerosol extinction profiles.

Purpose of the Study:

To address the debate on the uniqueness of two-frequency lidar inversion solutions.
To provide an analytical method for retrieving aerosol extinction profiles.

Main Methods:

Revisiting the two-frequency lidar inversion technique.
Developing an analytical solution for the lidar equation.

Main Results:

A unique solution for the two-frequency lidar inversion is shown to exist.
An analytical expression for the boundary value (aerosol transmission) and extinction profile is derived.

Conclusions:

The two-frequency lidar inversion can yield unique and analytical solutions.
This method advances the accurate retrieval of atmospheric aerosol properties.