Georeferencing GOES and parallax (Capstone 3 week)

Track 3 closes here. By Week 14 you can threshold-detect a hotspot in Band 7 brightness temperature. But the (x, y) pixel coordinate is not the (lat, lon) of the heat source on the ground — not directly. To get a usable geocoded detection, the pixel must be projected through the GOES fixed-grid math, and parallax must be corrected for tall plumes. That math is this week, and the capstone is a working end-to-end plume detector.

The GOES fixed grid

GOES doesn't deliver imagery in lat/lon. It delivers imagery in a scan-angle coordinate system tied to the satellite's frame of reference. Each pixel has an "x" value (east-west scan angle from satellite, in radians) and a "y" value (north-south elevation angle from satellite, in radians). The NetCDF file contains x and y 1-D coordinate arrays plus a goes_imager_projection variable with the geostationary projection metadata (perspective point altitude, satellite longitude, sweep angle axis).

To get the (lat, lon) of a pixel, you project a line from the satellite through the pixel down to the Earth ellipsoid:

import numpy as np
H = 35786023.0 + 6378137.0  # satellite altitude + Earth equatorial radius
λ0 = -137.2 * π / 180        # satellite longitude (GOES-18 West)
r_eq, r_pol = 6378137.0, 6356752.31414

x, y = scan_x, scan_y         # radians from the NetCDF

a = sin(x)**2 + (cos(x)**2 * (cos(y)**2 + (r_eq/r_pol)**2 * sin(y)**2))
b = -2 * H * cos(x) * cos(y)
c = H**2 - r_eq**2
r_s = (-b - sqrt(b**2 - 4*a*c)) / (2*a)

s_x = r_s * cos(x) * cos(y)
s_y = -r_s * sin(x)
s_z = r_s * cos(x) * sin(y)

lat = atan2((r_eq/r_pol)**2 * s_z, sqrt((H - s_x)**2 + s_y**2))
lon = λ0 + atan2(s_y, H - s_x)

This is the exact algorithm in the GOES-R Product Definition Document. satpy wraps it cleanly.

The parallax problem

The math above assumes the pixel lies on Earth's surface. A rocket plume at 50 km altitude is well above the surface, and from GOES's geostationary perspective, the apparent position of the plume is offset from the actual position of the rocket. This offset is the parallax.

For a plume directly below GOES (sub-satellite point), parallax is zero. Move 1,000 km north and parallax of a 50 km plume reaches ~5 km. Move to 4,000 km from the sub-satellite point and parallax is ~25 km — well above LaunchDetect's 5 km accuracy target.

Parallax correction: given the apparent (lat, lon) and an estimated plume altitude, compute the offset vector pointing back toward the sub-satellite point, scaled by altitude/(altitude + Earth radius). The corrected position is the apparent position shifted by that vector.

Limb effects

At the edge of GOES's visible disk (the "limb"), pixel geometry is highly oblique. A pixel that's nominally 2 km × 2 km in the center of the disk becomes 6+ km × 2 km near the limb (foreshortening), and the radiometric path length through the atmosphere is much longer (more attenuation and scatter). Practical rule: discard detections more than ~70° away from sub-satellite point.

The capstone

Week 15's lab is the start of Capstone 3: Thermal Plume Detector. Given a GOES Band 7 NetCDF for a known launch event, output records with (timestamp, lat, lon, brightness_temp_K, area_km²) for each plume detection, with parallax correction applied, FIRMS-based false-positive filtering, and a Folium heatmap visualization. The full rubric is on the capstone page; finishing it earns the Certified Remote Sensing Specialist credential. Track 4 (Mission GIS Engineer) starts next week, moving the focus into web-based delivery and 3D globes.