A fast piecewise 3D ray tracing algorithm for determining slant total delays

Abstract

Due to high computation demand, slant total delays (STDs) determined by 3D ray tracing cannot be widely used in geodetic analysis. This study proposes a novel fast 3D ray tracing approach, named APM-3D, based on piecewise algorithm and strictly matched step size. It can produce more than 1000 rays per second with a single core PC, which satisfies the operational applications of slant delays in mapping function establishment and other areas. Comparisons have shown mm-level agreement in slant delays at 5° elevation angle between APM-3D and several algorithms presented by different institutes in a comparison campaign. APM-3D also exhibits better stability than the standard RK4 method. Further validation using a 3D ray tracing based on the Cartesian coordinate system and 2D ray tracing also indicates excellent agreement, while constant differences in geometric delays between 3 and 2D exist under complex weather conditions. With ERA5 reanalysis data, the characteristics of STD evolution are presented during a rainstorm that occurred on July 1–2, 2020, over Wuhan.

Appendix A

To solve the equation $f\left( z \right) \equiv raytracing\left( z \right) - z_{e} = 0$, where z is the initial zenith angle of the ray, $z_{e}$ is the zenith angle from start point to end point, the secant method can be applied:

$$ z_{k + 1} - z_{k} = - f\left( {z_{k} } \right)/f^{\prime}\left( {z_{k} } \right) $$

(12)

where

$$ f^{\prime}\left( {z_{k} } \right) = \frac{{f\left( {z_{k} } \right) - f\left( {z_{k - 1} } \right)}}{{z_{k} - z_{k - 1} }} = \left( {1 - \frac{{f\left( {z_{k} } \right)}}{{f\left( {z_{k - 1} } \right)}}} \right)f{^{\prime}}\left( {z_{k - 1} } \right). $$

(13)

An empirical parameter factor can be introduced to reduce vibration in the procedure of iteration:

$$ f^{\prime}\left( {z_{k} } \right) = \left( {1 - factor{*}\frac{{f\left( {z_{k} } \right)}}{{f\left( {z_{k - 1} } \right)}}} \right)f{^{\prime}}\left( {z_{k - 1} } \right). $$

(14)

After a few numerical experiments, factor=0.970 is chosen in this study. The initial values are set as:

$$ z_{0} = z_{e} , f{^{\prime}}\left( {z_{0} } \right) = 1. $$

According to our analysis, this modified secant method can make the iteration procedure of raytracing converge quickly with high accuracy.