Tropo (Tropospheric Corrections)

Tropospheric delay and angular correction calculations based on ITU-R 2019 methods and ERA5 reanalysis data. Requires Python 3.12+.

Overview

gri-tropo provides tools for calculating tropospheric corrections and accessing atmospheric refractivity data:

• dgs: Calculate tropospheric angle and time corrections using ITU-R 2019 methods
• rates: Time derivatives of those corrections, for Doppler, FDOA, and TDOA-rate observables
• nsur: Access surface refractivity data from ERA5 reanalysis
• refht: Access reference height data (scale height of exponential refractivity decay)

Mathematical Background

The troposphere (0-~12 km altitude) causes signal delay and angular bending due to atmospheric refractivity. Unlike the ionosphere, tropospheric refraction is non-dispersive (frequency-independent) and affects all electromagnetic signals equally.

The refractivity profile is modeled as an exponential decay:

N(h) = N_sur * exp(-h / h_ref)

where N_sur is the surface refractivity in N-units, h is altitude in km, and h_ref is the reference (scale) height in km. The signal delay and angular bending are computed by integrating the refractivity along the propagation path using ray-tracing methods.

N_sur values typically range from 250-400 N-units and vary with latitude, season, and local weather. The reference height h_ref is typically 6-8 km.

Corrections over time

Rate observables are biased by the rate at which the troposphere lengthens the optical path, not by the delay itself. Differentiating the path with moving endpoints gives:

f_rx = f_tx * (1 - range_rate / c - d(tau)/dt)

so the whole tropospheric frequency term is -f_tx * d(tau)/dt, where tau is the one-way excess delay. Both the ray bending at each endpoint and the reduced propagation speed are already contained in tau, which is why dgs_tropo_rates differentiates the delay model directly rather than rotating the line of sight by the bending angle and re-differencing the geometric range. That geometric approach captures only one endpoint's bending and omits the speed term.

Because the troposphere is non-dispersive, the delay rate is frequency-independent and is carried in seconds per second, matching gri_utils.observables.get_tdoa_dot. The carrier scaling belongs at the point of use.

Model limits

The ITU-R 2019 fit is valid over a bounded range, and both entry points enforce the same limits:

• Emitter elevation is floored at -2 degrees.
• The time correction is clamped to [0, 500] ns.

dgs_tropo_rates differentiates the clamped model rather than the raw polynomials. Where the time correction is held at a limit, the delay rate is zero. Below the elevation floor the elevation dependence drops out of both rates, but neither goes to zero: the cone rate still follows the changing range, and both still follow the emitter altitude through the multiplicative correction.

References:

• ITU-R P.453. "The radio refractive index: its formula and refractivity data."
• ITU-R P.834. "Effects of tropospheric refraction on radiowave propagation."
• Bean & Dutton (1966). "Radio Meteorology." National Bureau of Standards Monograph 92.

Installation

pip install gri-tropo

For development:

git clone https://gitlab.com/geosol-foss/python/gri-tropo.git
cd gri-tropo
uv sync

Usage

DGS - Tropospheric Corrections

Calculate tropospheric angle and time corrections for signal propagation:

from gri_tropo.dgs import dgs_tropo_corrections
import numpy as np

# Define emitter location [lat, lon, alt_m]
emitter_lla = np.array([40.0, -105.0, 1500.0])

# Define collector position in ECEF XYZ (meters)
collector_xyz = np.array([1000000.0, -5000000.0, 4000000.0])

# Calculate corrections with custom atmospheric parameters
cone_corr, time_corr = dgs_tropo_corrections(
    emitter_lla,
    collector_xyz,
    n_sur=315.0,    # Surface refractivity (N-units)
    ref_ht=7.35     # Reference height (km)
)

# cone_corr: Angular deflection correction (radians)
# time_corr: Time delay correction (seconds)

Both outputs are one-way, single-path corrections for one emitter-to-collector leg. There is no built-in differential helper: for a TDOA, compute the time correction for each leg and subtract; for an AOA, use the cone correction directly. A single (3,) emitter returns floats, a (K, 3) batch returns two shape-(K,) arrays, and the batch axis is over emitters with one shared collector per call.

The n_sur and ref_ht defaults are global means. For location- and time-specific values, use the NSur and RefHt classes below.

DGS Rates - Tropospheric Corrections Over Time

Rate observables (Doppler, FDOA, TDOA-rate) are biased by the time derivative of the excess delay, not by the delay itself. dgs_tropo_rates returns the exact derivatives of the pair above for the same geometry:

from gri_tropo import dgs_tropo_rates
import numpy as np

emitter_lla = np.array([40.0, -105.0, 1500.0])
emitter_vel = np.zeros(3)                       # ECEF m/s
collector_xyz = np.array([1000000.0, -5000000.0, 4000000.0])
collector_vel = np.array([0.0, 7000.0, 1500.0])  # ECEF m/s

cone_rate, delay_rate = dgs_tropo_rates(
    emitter_lla,
    emitter_vel,
    collector_xyz,
    collector_vel,
)

# cone_rate:  radians per second
# delay_rate: seconds per second (dimensionless, frequency-independent)

The delay rate is frequency-independent because the troposphere is non-dispersive at RF, matching the convention of gri_utils.observables.get_tdoa_dot. Scale it by the carrier where it is used:

freq_bias_hz = -1.5e9 * delay_rate      # one-way bias at L-band

For an FDOA, compute the bias on each leg and subtract; the delay rate adds directly to a TDOA-rate observable.

NSur - Surface Refractivity Data

Access hourly surface refractivity data from ERA5:

from gri_tropo.nsur import NSur
from datetime import datetime

# Initialize with data directory
nsur = NSur("./data/n_sur")

# Get value for specific location and time
value = nsur.get_n_sur(
    lat=40.0,
    lon=254.0,  # Longitude 0-359 degrees East
    dt=datetime(2025, 3, 15),
    hour=12
)

# Get values for multiple coordinates efficiently
coords = [
    (40.0, 254.0, datetime(2025, 3, 15), 12),
    (35.0, 250.0, datetime(2025, 3, 15), 14),
]
values = nsur.get_all_n_surs(coords)

RefHt - Reference Height Data

Access reference height data (scale height for exponential refractivity decay):

from gri_tropo.refht import RefHt
from datetime import datetime

# Initialize with data directory
refht = RefHt("./data/ref_ht")

# Get value for specific location and date
value = refht.get_ref_ht(
    lat=40.0,
    lon=254.0,  # Longitude 0-359 degrees East
    dt=datetime(2025, 3, 15)
)

# Get values for multiple coordinates efficiently
coords = [
    (40.0, 254.0, datetime(2025, 3, 15)),
    (35.0, 250.0, datetime(2025, 3, 16)),
]
values = refht.get_all_ref_hts(coords)

Data Download and Processing

The NPZ data files required by this package are generated by the separate gri-tropo-data package. This separation keeps gri-tropo lightweight with minimal dependencies for users who just need tropospheric corrections.

For information on downloading ERA5 data and processing it to generate n_sur and ref_ht datasets, see the gri-tropo-data repository.

Dependencies

• gri-utils: Coordinate conversions and geodetic utilities
• numpy: Array operations
• scipy: Scientific computing

Other Projects

Current list of other GRI FOSS Projects we are building and maintaining.

License

MIT License. See LICENSE for details.