Beam Geometry

Beam Height Calculation

Daisho.jl calculates radar beam height using the standard 4/3 effective Earth radius model, which accounts for atmospheric refraction under standard conditions.

Effective Earth Radius

The effective Earth radius for standard refraction is:

\[R_{eff} = \frac{4}{3} R_E = \frac{4}{3} \times 6371000 \text{ m} \approx 8495333 \text{ m}\]

Beam Height Formula

The height of the radar beam center at slant range $r$ and elevation angle $\theta_e$ is:

\[h = \sqrt{r^2 + R_{eff}^2 + 2 r R_{eff} \sin(\theta_e)} - R_{eff} + h_{radar}\]

where $h_{radar}$ is the radar altitude above sea level.

This is implemented in Daisho.beam_height.

Reference

Doviak, R. J., and D. S. Zrnic, 1993: Doppler Radar and Weather Observations, 2nd ed., Academic Press.

Surface Range

The surface (ground) range corresponding to a beam at slant range $r$ and height $h$ is:

\[s = R_{eff} \arcsin\left(\frac{r \cos(\theta_e)}{R_{eff} + h}\right)\]

Coordinate System

Daisho uses a Transverse Mercator projection centered on the radar location (or a reference point). Coordinates are:

• Z: Height above sea level (meters)
• Y: Northward distance from the projection origin (meters)
• X: Eastward distance from the projection origin (meters)

The projection is computed using CoordRefSystems.jl for accuracy.

Approximate Inverse Projection

For converting projected (Y, X) coordinates back to (latitude, longitude), Daisho uses an approximate formula based on FCC wireless communication specifications:

\[\Delta\text{lat} = \frac{Y}{f_{lat} \times 1000}\]

\[\Delta\text{lon} = \frac{X}{f_{lon} \times 1000}\]

where the latitude and longitude factors are:

\[f_{lat} = 111.13209 - 0.56605 \cos(2\phi) + 0.00012 \cos(4\phi) - 0.000002 \cos(6\phi)\]

\[f_{lon} = 111.41513 \cos(\phi) - 0.09455 \cos(3\phi) + 0.00012 \cos(5\phi)\]

This is implemented in Daisho.appx_inverse_projection.