Setting the Surface BRDF

import sasktran2 as sk

SASKTRAN2 has the capability to include surface reflectance in the form of a Bi-directional Reflectance Distribution Function (BRDF). The simplest BRDF is a Lambertian surface where the outgoing radiation is the same in all directions.

Let’s start by setting up a calculation where we view the ground at a variety of different viewing zenith angles.

import numpy as np
import matplotlib.pyplot as plt

config = sk.Config()
#config.multiple_scatter_source = sk.MultipleScatterSource.DiscreteOrdinates
config.num_streams = 4

model_geometry = sk.Geometry1D(cos_sza=0.6,
                                solar_azimuth=0,
                                earth_radius_m=6372000,
                                altitude_grid_m=np.arange(0, 65001, 1000),
                                interpolation_method=sk.InterpolationMethod.LinearInterpolation,
                                geometry_type=sk.GeometryType.Spherical)

viewing_geo = sk.ViewingGeometry()

cos_viewing_angle = np.arange(0.01, 0.99, 0.05)

for mu in cos_viewing_angle:
    ray = sk.GroundViewingSolar(0.6, np.pi, mu, 100000)
    viewing_geo.add_ray(ray)

for mu in cos_viewing_angle[::-1]:
    ray = sk.GroundViewingSolar(0.6, 0, mu, 100000)
    viewing_geo.add_ray(ray)

wavel = np.array([600])

atmosphere = sk.Atmosphere(model_geometry, config, wavelengths_nm=wavel)

sk.climatology.us76.add_us76_standard_atmosphere(atmosphere)

atmosphere['rayleigh'] = sk.constituent.Rayleigh()
atmosphere['ozone'] = sk.climatology.mipas.constituent("O3", sk.optical.O3DBM())

engine = sk.Engine(config, model_geometry, viewing_geo)

BRDFs are added to the atmosphere the same way any constituent is. Let’s add a Lambertian surface with a constant surface albedo of 1.0

atmosphere['brdf'] = sk.constituent.LambertianSurface(1.0)

And calculate the radiance

radiance = engine.calculate_radiance(atmosphere)

radiance["radiance"].isel(stokes=0).sel(wavelength=600).plot()

[<matplotlib.lines.Line2D at 0x7b34f972c890>]

We can change the surface to a snow surface and compare the results,

atmosphere['brdf'] = sk.constituent.SnowKokhanovsky()

radiance_snow = engine.calculate_radiance(atmosphere)

radiance["radiance"].isel(stokes=0).sel(wavelength=600).plot()
radiance_snow["radiance"].isel(stokes=0).sel(wavelength=600).plot()

[<matplotlib.lines.Line2D at 0x7b34f97bacf0>]

And we see a shifting of radiance values towards the later lines of sight, which have relative azimuth 0 and are in the glint region.

Surface BRDF Linearization

Each BRDF provides linearizations with respect to it’s input parameters. If we look at the radiance calculated with the Lambertian surface,

print(radiance)

<xarray.Dataset> Size: 80kB
Dimensions:               (wavelength: 1, los: 40, stokes: 1, altitude: 66,
                           ozone_altitude: 50, brdf_wavelength: 1)
Coordinates:
  * wavelength            (wavelength) float64 8B 600.0
  * stokes                (stokes) <U1 4B 'I'
Dimensions without coordinates: los, altitude, ozone_altitude, brdf_wavelength
Data variables:
    radiance              (wavelength, los, stokes) float64 320B 0.05523 ... ...
    wf_temperature_k      (altitude, wavelength, los, stokes) float64 21kB 6....
    wf_ozone_vmr          (ozone_altitude, wavelength, los, stokes) float64 16kB ...
    wf_pressure_pa        (altitude, wavelength, los, stokes) float64 21kB -1...
    wf_specific_humidity  (altitude, wavelength, los, stokes) float64 21kB 0....
    wf_brdf_albedo        (brdf_wavelength, wavelength, los, stokes) float64 320B ...

We see there is a field wf_brdf_albedo that represents the derivative of the observations with respect to our input albedo value.

Looking at the snow calculation,

print(radiance_snow)

<xarray.Dataset> Size: 81kB
Dimensions:               (wavelength: 1, los: 40, stokes: 1, altitude: 66,
                           ozone_altitude: 50, brdf_wavelength: 1)
Coordinates:
  * wavelength            (wavelength) float64 8B 600.0
  * stokes                (stokes) <U1 4B 'I'
Dimensions without coordinates: los, altitude, ozone_altitude, brdf_wavelength
Data variables:
    radiance              (wavelength, los, stokes) float64 320B 0.05351 ... ...
    wf_temperature_k      (altitude, wavelength, los, stokes) float64 21kB 4....
    wf_pressure_pa        (altitude, wavelength, los, stokes) float64 21kB -1...
    wf_specific_humidity  (altitude, wavelength, los, stokes) float64 21kB 0....
    wf_ozone_vmr          (ozone_altitude, wavelength, los, stokes) float64 16kB ...
    wf_brdf_L             (brdf_wavelength, wavelength, los, stokes) float64 320B ...
    wf_brdf_albedo        (brdf_wavelength, wavelength, los, stokes) float64 320B ...
    wf_brdf_M             (brdf_wavelength, wavelength, los, stokes) float64 320B ...

We see two parameters, wf_brdf_L and wf_brdf_M. These are the derivatives with respect to the input parameters L and M of the Kokhanovsky model.

Available BRDFs

A full list of available surface parameterizations can be found at BRDFs.