pyi2em

Python bindings for the I2EM rough-surface scattering and emissivity model.

Reference: Ulaby, F.T. and Long, D.G. (2014). Microwave Radar and Radiometric Remote Sensing. University of Michigan Press.

Install

pip install pyi2em

Notes on wheels vs source builds

• PyPI wheels bundle required native libraries, including GSL and embedded cubature routines. End users do not need system packages.
• Source builds require GSL available via pkg-config. The cubature integration code is compiled into the extension automatically.

From source

• Ubuntu: sudo apt-get install -y cmake build-essential libgsl-dev
• macOS: brew install cmake gsl

Then

pip install .

Note: The cubature library (hcubature/pcubature) is fetched automatically during build; you do not need to install a system package for it. GSL remains a system dependency for source builds.

Development with pixi (recommended for source builds)

pixi provides a fully reproducible build/dev environment from the lockfile — including the GSL and C++ toolchain — so you do not need to install any system packages (no apt/brew). The environment is defined in the [tool.pixi.*] tables of pyproject.toml; pixi.lock pins exact versions.

# Install pixi (one time): https://pixi.sh/latest/#installation
pixi install        # resolves toolchain + GSL, builds pyi2em editable
pixi run test       # run the test suite
pixi run validate   # run the acceptance checks
pixi run example    # run example.py

After editing the C++ sources, force a fresh native recompile:

pixi run rebuild

Test

pytest -q

Or, inside the pixi environment: pixi run test.

Usage

The high-level Python API exposes three functions. Units are GHz, meters, and degrees throughout.

1. Emissivity

from pyi2em import emissivity

eh, ev = emissivity(
    freq_ghz=5.3,
    rms_height_m=0.0025,
    corr_length_m=0.10,
    theta_deg=30.0,
    er_complex=complex(11.3, 1.5),
    correl="gaussian",         # 'exponential' | 'gaussian' | 'x_power' | 'x_exponential'
)
print(eh, ev)  # linear emissivities

2. Monostatic backscatter (scalar or array incidence)

import numpy as np
from pyi2em import sigma0_backscatter

thetas = np.array([20.0, 30.0, 40.0])
sig = sigma0_backscatter(
    freq_ghz=5.0,
    rms_height_m=0.01,
    corr_length_m=0.05,
    theta_deg=thetas,                 # scalar or array
    er_complex=complex(15.0, 3.0),
    correl="gaussian",
    include_hv=True,
    return_db=True,                   # return dB (default)
)
print(sig["hh"], sig["vv"], sig["hv"])  # arrays in dB

3. Bistatic backscatter

from pyi2em import sigma0_bistatic

sig = sigma0_bistatic(
    freq_ghz=5.0,
    rms_height_m=0.01,
    corr_length_m=0.05,
    thi_deg=40.0,
    ths_deg=40.0,
    phs_deg=180.0,
    er_complex=complex(15.0, 3.0),
    correl="gaussian",
    xcoeff=1.0,
    return_db=True,
)
print(sig["hh"], sig["vv"])  # dB

Correlation options: 'exponential' | 'gaussian' | 'x_power' | 'x_exponential'.

Notes

• sigma0_* return dB by default; set return_db=False for linear values.
• sigma0_backscatter(theta_deg=...) accepts a scalar or a NumPy array.
• emissivity returns linear values in [0, 1].

Validation

The cross-pol (VH) channel is validated against the published reference from Fung and Chen, Microwave Scattering and Emission Models for Users (Artech House, 2010), Fig. 3.9 — dielectric variation with exponential correlation (f = 5 GHz, σ = 0.3 cm, L = 4 cm). pyi2em reproduces the reference curves to within ~0.3 dB across incidence angles 0–70° and ε_r = 3.5 / 7.5 / 15.

Regenerate with python docs/make_xpol_figure.py; the numerical check lives in tests/test_xpol_reference.py.

Bistatic (cross-model)

The bistatic model is referenced to the IEM-M single-surface bistatic program (Fung and Chen, 2010, Figs 6.1/6.2 — azimuthal scattering at f = 10 GHz, θ_i = 20°, θ_s = 50°, ε_r = 3, L = 3 cm, exponential). pyi2em implements I2EM (Ulaby & Long 2014) and Fung's reference is IEM-M so this is a labeled cross-model comparison. Away from the specular null the two agree to a median ~0.2 dB across azimuth for σ = 0.2/0.4/0.7 cm. Divergence is limited to a narrow specular-null notch, indicating a small null-position shift between the formulations.

Regenerate with python docs/make_bistatic_figure.py.

Reference

Fung, A.K. and Chen, K.S.(2010), Microwave Scattering and Emission Models for Users, Artech House
Ulaby, F.T. and Long, D.G.(2014), Microwave Radar and Radiometric Remote Sensing, The University of Michigan Press

License and Notices

• Wrapper: MIT (see LICENSE)
• GSL: GPL-licensed dependency; the NOTICE and COPYING.GPL files are included.