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The unit circle—a circle with a radius of one, centered at the origin of the coordinate plane—serves as the foundational framework for defining trigonometric functions. In this context, arc length refers to the distance measured along the circumference of the circle between two points, and it provides a way to represent real numbers geometrically. Each real number t corresponds to an arc length measured counterclockwise from the positive x-axis around the circle. The coordinates of...
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Iterative discrete ordinates solution of the equation for the surface reflected radiance Alexander Radkevich.

Journal of quantitative spectroscopy & radiative transfer·2020
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Updated: Jan 5, 2026

Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator
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Modified Geometric Truncation of the Scattering Phase Function.

Alexander Radkevich1

  • 1Science Systems and Applications, Inc., 1 Enterprise Pkwy, Hampton, VA, USA, 23666.

Journal of Quantitative Spectroscopy & Radiative Transfer
|October 22, 2019
PubMed
Summary

Accurate light scattering calculations for remote sensing are improved by modifying the phase function. New methods enhance accuracy and computational efficiency for atmospheric particle scattering.

Keywords:
angle of truncationdelta-Mphase functionsimilarity transformationtruncation

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Area of Science:

  • Atmospheric Optics
  • Radiative Transfer Theory
  • Computational Physics

Background:

  • Light scattering by large atmospheric particles presents challenges for numerical radiance calculations due to strong forward peaks.
  • Accurate radiance computations are crucial for remote sensing applications.

Purpose of the Study:

  • To develop and evaluate modified phase functions for improved numerical calculations of light scattering.
  • To address the limitations of geometric truncation and propose objective methods for selecting truncation angles.

Main Methods:

  • Phase function modification using scaling transformation, including geometric truncation and a new continuous functional form.
  • Investigating modifications preserving asymmetry parameter, continuous first derivative, and mean scattering angle.
  • Developing a heuristic approach for objective selection of the forward cone width.
  • Testing modifications using discrete ordinates and Monte Carlo methods on cloud phase functions.

Main Results:

  • Modified phase functions, particularly those with continuous derivatives, enhance radiance computation accuracy compared to original geometric truncation.
  • The heuristic approach provides an unambiguous criterion for selecting the truncation angle.
  • Continuous derivative modifications offer significant computational time savings in Monte Carlo simulations.

Conclusions:

  • The proposed phase function modifications and objective truncation angle selection improve the accuracy and efficiency of radiative transfer calculations.
  • These advancements are beneficial for remote sensing applications relying on accurate atmospheric light scattering data.