Source code for radarx.fundamentals.scattering
"""
Rayleigh Scattering Approximations
==================================
Functions for computing backscatter cross-sections, size parameters,
and absorption, scattering, and extinction coefficients under Rayleigh
scattering assumptions.
.. autosummary::
:nosignatures:
:toctree: generated/
backscatter_cross_section
normalized_backscatter_cross_section
size_parameter
absorption_coefficient
scattering_coefficient
extinction_coefficient
References
----------
- Rinehart (1997). Radar for Meteorologists.
- Battan (1973). Radar Observations of the Atmosphere.
"""
import numpy as np
[docs]
def backscatter_cross_section(diameter, wavelength, dielectric=0.93):
"""
Rayleigh backscatter cross-section for a water sphere.
Parameters
----------
diameter : float or array-like
Drop diameter [m]
wavelength : float
Radar wavelength [m]
dielectric : float
Dielectric factor (default: 0.93 for water)
Returns
-------
float or array-like
Backscatter cross-section [m^2]
"""
d = np.asarray(diameter)
return (np.pi**5 * dielectric**2 * d**6) / wavelength**4
[docs]
def normalized_backscatter_cross_section(diameter, wavelength, dielectric=0.93):
"""
Normalize Rayleigh backscatter cross-section by projected area.
Parameters
----------
diameter : float or array-like
Drop diameter [m]
wavelength : float
Radar wavelength [m]
dielectric : float, optional
Dielectric factor (default: 0.93)
Returns
-------
float or array-like
Normalized cross-section [unitless]
"""
sigma = backscatter_cross_section(diameter, wavelength, dielectric)
area = np.pi * (np.asarray(diameter) / 2.0) ** 2
return sigma / area
[docs]
def size_parameter(diameter, wavelength):
"""
Calculate size parameter alpha = π * D / λ.
Parameters
----------
diameter : float or array-like
Drop diameter [m]
wavelength : float
Radar wavelength [m]
Returns
-------
float or array-like
Size parameter (unitless)
"""
return np.pi * np.asarray(diameter) / wavelength
def _size_parameter(radius, wavelength):
"""
Calculate size parameter x = 2π * radius / wavelength.
Parameters
----------
radius : float or array-like
Particle radius [m]
wavelength : float
Radar wavelength [m]
Returns
-------
float or array-like
Size parameter (unitless)
"""
return 2 * np.pi * np.asarray(radius) / wavelength
def _complex_ratio(refractive_index):
"""
Calculate the complex ratio (m^2 - 1) / (m^2 + 2).
Parameters
----------
refractive_index : complex
Complex refractive index of the particle
Returns
-------
complex or array-like of complex
Complex ratio used in scattering calculations
"""
m = refractive_index
m2 = m * m
return (m2 - 1) / (m2 + 2)
[docs]
def absorption_coefficient(radius, wavelength, refractive_index):
"""
Compute Rayleigh absorption coefficient.
Parameters
----------
radius : float or array-like
Particle radius [m]
wavelength : float
Radar wavelength [m]
refractive_index : complex
Complex refractive index of the particle
Returns
-------
float or array-like
Absorption efficiency (Qa)
"""
x = 2 * np.pi * radius / wavelength
m = refractive_index
m2 = m * m
return np.maximum(4 * x * np.imag((m2 - 1) / (m2 + 2)), 0.0)
[docs]
def scattering_coefficient(radius, wavelength, refractive_index):
"""
Compute Rayleigh scattering coefficient.
Parameters
----------
radius : float or array-like
Particle radius [m]
wavelength : float
Radar wavelength [m]
refractive_index : complex
Complex refractive index of the particle
Returns
-------
float or array-like
Scattering efficiency (Qs)
"""
x = _size_parameter(radius, wavelength)
ratio = _complex_ratio(refractive_index)
return (8 / 3) * x**4 * np.abs(ratio) ** 2
[docs]
def extinction_coefficient(radius, wavelength, refractive_index):
"""
Compute Rayleigh extinction coefficient (Qa + Qs).
Parameters
----------
radius : float or array-like
Particle radius [m]
wavelength : float
Radar wavelength [m]
refractive_index : complex
Complex refractive index of the particle
Returns
-------
float or array-like
Extinction efficiency (Qe)
"""
return np.maximum(
absorption_coefficient(radius, wavelength, refractive_index)
+ scattering_coefficient(radius, wavelength, refractive_index),
0.0,
)
__all__ = [
"backscatter_cross_section",
"normalized_backscatter_cross_section",
"size_parameter",
"absorption_coefficient",
"scattering_coefficient",
"extinction_coefficient",
]
