28 - Microwave principles and parameters

This chapter is for readers who know optical remote sensing and are new to microwave simulation: which parameters a microwave simulation needs, how to choose scattering models, and how to read the results. LESS has two microwave sensors: RadarScatterometer computes polarimetric backscattering coefficients (chapter 29) and MicrowaveRadiometer computes polarised brightness temperatures (chapter 30).

The parameters in the examples illustrate the method; they are not calibrated values for any species or soil.

From optical to microwave

Optical concept	Microwave counterpart	Difference
Wavelength, spectrum	Frequency frequency (Hz)	The wavelength is often comparable to leaf and branch sizes
Leaf reflectance, transmittance	Complex permittivity and scatterer model	Shape, size, water content and orientation together determine scattering
Leaf angle distribution, leaf area index	Number density, size distribution and orientation distribution of scatterers	The same leaf area index can give different microwave responses
Bidirectional reflectance, phase function	Polarimetric scattering matrix	Conversion between polarisations must also be computed
Optical depth	Polarisation-dependent extinction and optical depth	Changes with frequency and propagation direction
Reflection of sunlight	Backscatter of radar illumination	Transmit and receive directions coincide
Thermal emission and temperature	Microwave emission and brightness temperature	Brightness temperature depends on temperature, emissivity, background and canopy together

In the optical domain a leaf is much larger than the wavelength; at microwave frequencies it can be smaller than or comparable to the wavelength, and then leaf size and the internal electric field affect scattering.

Frequency and scale

\[\lambda = c/f, \qquad k = 2\pi/\lambda\]

Frequency	Wavelength	Notes
P band, 435 MHz	69 cm	Leaves are relatively small; large branches and trunks dominate
L band, 1.4 GHz	21.4 cm	Widely used for soil moisture and vegetation
C band, 5.4 GHz	5.6 cm	Centimetre-sized leaves and branches are no longer small particles
X band, 9.6 GHz	3.1 cm	Fine structures and the upper canopy matter more

Penetration depth also depends on water content, scatterer density, orientation and incidence angle. When the same forest is observed at another frequency, the dielectric properties, scattering and canopy extinction must all be recomputed.

What active and passive sensors measure

Backscattering coefficients

sigma0_HH, sigma0_HV, sigma0_VH and sigma0_VV are linear, dimensionless backscattering coefficients. H and V are horizontal and vertical polarisations relative to the observation reference plane; HH and VV are co-polarised and HV and VH cross-polarised. Cross-polarisation means the polarisation direction has been converted, which can happen in a single scattering event on tilted non-spherical scatterers.

Convert to decibels with 10 log₁₀ σ⁰. Average spatially on linear values first and then convert to decibels. Backscattering coefficients are not reflectances and can exceed 1.

Brightness temperature

For an opaque smooth surface at uniform temperature:

\[T_{B,p} = e_p\,T_s + (1 - e_p)\,T_{\mathrm{down},p}\]

Tₛ is the physical temperature, eₚ the emissivity at that polarisation and direction, and the second term is the reflected sky background. A surface at 300 K with emissivity 0.8, for example, has a brightness temperature of about 240 K without background; this does not mean that the soil is at 240 K.

A canopy attenuates soil emission and also emits and scatters itself, and the atmosphere likewise attenuates and emits, so the soil temperature cannot be read directly from the brightness temperature of a forest. The SMAP passive microwave algorithm document describes this radiation budget and common simplified canopy models.

Emissivity follows from absorption by Kirchhoff's law. For rough soil, the total reflection into all directions for a given incident direction is needed; a single backscattering coefficient cannot replace it.

How LESS computes

1. Materials: complex permittivity from frequency, water content, dry matter density and so on;
2. Single scatterers: polarimetric scattering of discs or cylinders, with an approximation chosen by size;
3. Scatterer populations: averaging over orientation distributions and combining with number density gives direction-dependent extinction, scattering, absorption and polarisation conversion;
4. 3-D propagation: radiation travels through crown volumes, trunks and terrain, with obstruction, attenuation and scattering;
5. Observation: in active mode the contributions returning towards the radar are accumulated; in passive mode the contributions from soil, crowns, trunks, sky and atmosphere.

For sparse, independent scatterers, the extinction coefficient and optical depth can be written

\[\beta_{\mathrm{ext}} = \sum_j n_j \langle C_{\mathrm{ext},j} \rangle, \qquad \tau = \int \beta_{\mathrm{ext}}\,ds\]

nⱼ is number density (m⁻³) and C the cross-section of one scatterer (m²), so the extinction coefficient is in m⁻¹. The polarimetric propagation is more complete than this scalar formula, but the idea is the same.

scattering_samples sets the number of samples for angular integrals over scatterers, and quality the number of propagation samples per pixel. Neither is the number of real leaves in the scene. The output is a statistical radiative transfer result and does not include the pulse-by-pulse complex echoes needed for synthetic aperture radar imaging.

Describing vegetation

Crowns, scatterers and trunks

• Crown boundary: TurbidBoundary gives a closed volume where the scatterers are; the boundary itself does not scatter;
• Scatterers in the crown: MicrowaveVegetation(components=[...]) describes the statistical make-up of leaves, needles and branches;
• Trunks: a triangle mesh with MicrowaveTrunk; scattering is computed for an equivalent cylinder, while the mesh provides position and obstruction.

A canopy can be split into several volumes with their own density and water content to distinguish layers or species. Overlapping volumes add up, so make sure the same biomass is not counted twice.

Parameters

Parameter	Meaning and unit	Data source
radius	Radius of disc or cylinder (m)	Leaf area, needle or branch diameter
thickness	Leaf thickness (m)	Leaf thickness measurements
length	Needle or branch length (m)	Length statistics by size class
density	Number density (m⁻³)	Total number of scatterers divided by the volume they occupy
moisture	Gravimetric water content on a wet basis (0–1)	Fresh and oven-dry weights
dry_density	Dry matter density in the dielectric model (g cm⁻³)	Measurements consistent with the dielectric model used
orientation	Orientation distribution of leaf normals or cylinder axes	Leaf and branch angle measurements
temperature	Physical temperature (K)	Temperature measurements, especially for passive simulations

Wet-basis water content M_g = (fresh − dry) / fresh; for 10 g fresh and 4 g dry weight it is 0.6. If the data are dry-basis water content u = (fresh − dry) / dry, convert first with M_g = u / (1 + u). Neither the PROSPECT equivalent water thickness nor the volumetric water content of soil belongs here.

dry_density is entirely different from the number density density. When dry_density > 0, the gravimetric water content is converted to volumetric water content with M_v = M_g ρ_d / [1 − M_g (1 − ρ_d)] for the dielectric model; with dry_density = 0 a dielectric model based on gravimetric water content is used directly. These are two different dielectric models, and 0 does not mean that there is no dry matter.

Leaf number density from leaf area index

A disc leaf of radius r has a one-sided area of πr², so

\[\mathrm{LAD} = n_l\,\pi r^2, \qquad n_l = \frac{\mathrm{LAI}\,A_g}{\pi r^2\,V_c}\]

A_g is the ground area that the leaf area index refers to, and V_c the total crown volume containing those leaves. For a uniform canopy of thickness H covering the ground, n_l = LAI / (H πr²).

For leaf area index 3, a 5 m deep canopy and an equivalent leaf radius of 2 cm, the leaf area density is 0.6 m² m⁻³ and the number density about 477 m⁻³. For discrete crowns use the actual crown volume, not the tree height.

At the same leaf area, many small leaves and a few large leaves can respond differently at microwave frequencies. When leaf sizes differ clearly, use several MicrowaveLeaf components, each with its own number density.

Branches and needles

MicrowaveNeedle and MicrowaveBranch are both cylindrical scatterers. A cylinder has volume πr²L, so a group of cylinders with volume fraction φ has number density n = φ / (πr²L). To convert from dry biomass, first convert to volume with the dry matter density, then divide by the volume of one cylinder.

Large branches, twigs and needles are best set separately; one average cylinder cannot represent scattering over all sizes. Woody parts already described as trunks should not be counted again as branches in the crown.

Orientation

For leaves, orientation describes the angle between the leaf normal and the vertical; a horizontal leaf has a nearly vertical normal. For cylinders, orientation describes the angle between the axis and the vertical; a horizontal branch has a nearly horizontal axis.

"spherical" means directions uniformly distributed on the sphere, not inclination angles equally likely between 0° and 90°. "planophile" and "erectophile" correspond to those leaf angle types; Beta can fit a leaf angle distribution, and SinePower suits branch and needle axes. All these distributions are symmetric in azimuth.

Scattering models for leaves and branches

Rayleigh: scatterers that are electrically small enough, approximated as induced dipoles. Its validity depends not only on the ratio of size to wavelength but also on permittivity, since the wavelength inside a wet leaf is shorter than in air. Leaves are treated as oblate spheroids and cylinders as prolate spheroids of equal volume.

Physical optics (leaves): treats a thin leaf as a resistive sheet, suitable for thin leaves with significant in-plane size. It keeps the electromagnetic response of a finite sheet and is not a diffuse-reflection approximation.

Cylinder models: generalized_rayleigh_gans is for slender cylinders and requires the electrically thin condition; infinite_cylinder builds the response of a finite cylinder from the solution for an infinite one.

scattering_model="auto" chooses according to:

Shape	Rule
Disc leaf of diameter d	Physical optics when λ/d < 1.5, otherwise Rayleigh
Cylinder	First check the electrically thin condition kd|√εᵣ| ≤ 0.1
Electrically thin cylinder	Generalised Rayleigh–Gans for L ≥ 2λ, Rayleigh for L < 1.5λ
Other cylinders	Cylinder solution

These rules follow the scatterer model selection of MIMICS (MIMICS model and code). A 4 cm diameter leaf, for example, has λ/d ≈ 5.35 at 1.4 GHz and uses Rayleigh; at 5.4 GHz λ/d ≈ 1.39 and physical optics is used. For spectral analyses across a switching frequency, check that the results are continuous. More samples only reduce random error; they do not change the error of the scattering approximation itself.

Soil

For soil, a dielectric model and a rough-surface model are chosen separately.

Dielectric models

soil_moisture is volumetric water content (m³ m⁻³), and sand_fraction and clay_fraction are the sand and clay fractions (0–1). dielectric_model can be mironov, dobson or hallikainen. These are different empirical models; where possible, use the version, frequency range and soil type of your data or of the model you compare with. Check applicability separately for frozen or saline soils.

Rough-surface models

rms_height_cm is the standard deviation of surface height and correlation_length_cm the horizontal correlation length, both in centimetres, unlike the metres used for vegetation. The elevation model describes macroscopic slopes and roughness the smaller-scale undulations; do not include undulations of the same scale in both.

Before choosing a model, compute ks = 2πs/λ and kl = 2πl/λ and consider the scale of slopes and curvature. The small perturbation and physical optics models assume a Gaussian correlation function, and geometric optics uses the corresponding Gaussian slope statistics.

surface_model	Use	Conditions
specular	Smooth interface	Also set the height standard deviation to 0
small_perturbation	First-order small perturbation	Height variations small relative to the wavelength, small slopes
physical_optics	Kirchhoff physical optics	Locally approximable by tangent planes, large radius of curvature
geometric_optics	Geometric optics limit of Kirchhoff	Stationary-phase conditions at higher frequencies
oh	Oh empirical backscatter model	Active single scattering only; not for passive or multiple scattering
hqn	Empirical rough-surface emissivity model	Passive only; H, Q and N are model parameters, not roughness

First decide whether the surface satisfies the local tangent-plane approximation; if it does, use physical optics when the wave effects of the surface integral matter, and geometric optics only in the high-frequency stationary-phase range. With sharp undulations, small curvature radii or significant shadowing, these approximations may all fail. A numerical study of rough surfaces shows how curvature and roughness affect their accuracy.

For s = 0.8 cm and l = 6 cm, for example, ks ≈ 0.23 and kl ≈ 1.76 at 1.4 GHz, and about 0.91 and 6.79 at 5.4 GHz. The geometry is the same, but the electromagnetic roughness is clearly different.

The Oh model (Oh, Sarabandi & Ulaby, 1992) is empirical; before using it, check that its measurements cover the frequencies and roughness you are studying. physical_optics for leaves and for soil are two different models with different validity conditions.

Scattering order and trunks

Active mode uses scattering_order=1 by default, and the result is decomposed into crown, ground, trunk and combined paths. The order counts incoherent scattering events, so "ground specular reflection – crown scattering – ground specular reflection" is still first order. scattering_order=None accumulates up to max_scatter_depth orders; increase the maximum order and the number of samples separately to make sure the result is stable. The Oh model cannot provide the bistatic scattering needed for multiple scattering.

Passive mode always includes multiple scattering, and thermal emission follows from absorption. Explicit trunks should have diameters no smaller than the wavelength; thinner woody parts belong in the crown as MicrowaveBranch.

Example: from bare soil to a leafy crown

The code below runs in order: first bare soil, then a crown with leaves only. image_size=16, quality=16 and scattering_samples=8192 are for demonstration; quantitative work needs a convergence check.

Bare soil

import numpy as np
import less

scene = less.Scene()
scene.size = 30.0
scene.repetitive = True
scene.terrain = less.Terrain(property=less.MicrowaveSoil(
    dielectric_model="mironov", soil_moisture=0.20,
    sand_fraction=0.30, clay_fraction=0.20,
    surface_model="small_perturbation",
    rms_height_cm=0.3, correlation_length_cm=6.0,
    temperature_surface=295.0, temperature_deep=295.0))

geometry = less.Orthographic(image_size=16, view_zenith=35.0)
radar = less.RadarScatterometer(geometry=geometry, frequency=1.4e9,
                                quality=16, scattering_samples=8192)
radiometer = less.MicrowaveRadiometer(geometry=geometry, frequency=1.4e9, polarization="HV",
                                      quality=16, scattering_samples=8192,
                                      atmosphere_optical_depth=0.0)

bare_radar = scene.simulate(radar)
bare_tb = scene.simulate(radiometer)
print("bare soil HH:", np.mean(bare_radar.sigma0_HH))
print("bare soil H, V brightness temperature (K):", np.mean(bare_tb.Tb_H), np.mean(bare_tb.Tb_V))

With an atmospheric optical depth of 0 there is no atmospheric effect, but the cosmic background is still present.

Adding a leafy crown

crown = less.Object("leaf_crown")
crown.add_component("crown", less.TurbidBoundary(less.asset_path("tutorial_tree.obj"), group="crown"))
crown.set_property("crown", less.MicrowaveVegetation(components=[
    less.MicrowaveLeaf(radius=0.02, thickness=0.0002, density=477.0,
                       moisture=0.60, dry_density=0.30, temperature=298.0,
                       orientation="spherical", scattering_model="auto")]))
scene.add(crown, positions=[(15.0, 15.0, 0.0)])

forest_radar = scene.simulate(radar)
forest_tb = scene.simulate(radiometer)
print("with crown HH:", np.mean(forest_radar.sigma0_HH))
print("direct crown HH:", np.mean(forest_radar.mechanisms["direct_crown"].hh))
print("with crown H, V brightness temperature (K):", np.mean(forest_tb.Tb_H), np.mean(forest_tb.Tb_V))

477 m⁻³ corresponds to a leaf area density of about 0.6 m² m⁻³. There is only one crown in the scene, so the leaf area index of the 30 m × 30 m scene is far below 3, and the image mean also includes many bare-soil pixels.

Building up trustworthy results

1. Bare soil: first check Fresnel reflection and brightness temperature with a smooth interface, then add roughness. At oblique viewing, specular reflection from smooth ground does not return to the radar; weak backscatter does not mean the ground reflects little;
2. One scatterer population: add leaves, needles or one group of branches at a time and look at the effects of frequency, angle and density;
3. Change temperature and water content separately: changes in brightness temperature can come from temperature, or from dielectric properties, absorption and obstruction;
4. Add trunks and combined paths: look at the mechanism decomposition and check that woody parts are not counted twice;
5. Check convergence: increase scattering_samples and quality separately, and for active multiple scattering also max_scatter_depth, changing one thing at a time;
6. Compare with independent references: frequency, polarisation, geometry, soil and scatterer models, temperatures and boundary conditions must all match.

Wet soil usually reflects more and emits less, but the canopy and roughness can change the overall trend; more leaves increase scattering and emission but also obstruction, so backscatter and brightness temperature do not necessarily change monotonically.

Parameters to collect

• Observation: frequency, incidence angle, polarisation;
• soil: texture, volumetric water content, roughness and the scale at which it was measured, temperature;
• for each kind of leaf, needle and branch: size distribution, number density, wet-basis water content, dry matter density, orientation;
• crown volumes, trunk sizes and positions, terrain;
• the chosen dielectric and scattering models.

With only leaf area index, tree height and optical reflectance you can build exploratory scenes, but that is not enough to fix the microwave parameters. Add leaf sizes, branch size classes, water content and soil roughness first; for unknown parameters, give plausible ranges and run sensitivity analyses.

Next

• 29 - Radar backscatter
• 30 - Microwave brightness temperature