VLIDORT User's Guide

the models. Surface reflectance Jacobians have also been considered in other linearized RTmodels [Hasekamp and Landgraf, 2002; Ustinov, 2005].These kernels were developed for scalar BRDFs – the (1,1) component of the polarized surfacereflectance matrix. All scalar BRDFs have been implemented as part of the VLIDORT package.Polarized BRDFs over land surfaces are harder to come by. Here we report briefly on some newsemi-empirical formulae for BPDFs (Bidirectional Polarized Distribution Functions) [Maignan etal., 2009]. These BPDF kernels were supplied by F.-M. Bréon, and permission has been grantedto use them in VLIDORT, provided the work is properly acknowledged using the above 2009reference. See also section 3.2.4.6.3.6. Polarized land surface BRDF kernelsIn general, BPDFs are “spectrally neutral”, and modeling using specular reflection has becomethe accepted way of generating these functions. An empirical model was developed in part fromspecular reflection and in part from an analysis of POLDER measurements [Nada and Bréon,1999]. Recently, a great deal more BPDF information has been gleaned from data analysis ofseveral years of measurements from the PARASOL instrument. Based on this analysis, the paperof [Maignan et al., 2009] gives the following empirical formula for the BPDF:C exp[ − tanγ]exp[ −NDVI]Fp(γ , n)Rp(ΩS, ΩV) =; (6.3.14)μ + μSVHere, Ω S and Ω V are the incident and reflected geometries for nadir angles μ S and μ V , γ is half thephase angle of reflectance, n the refractive index of the vegetative matter (taken to be 1.5), C is aconstant, and Fp is the Fresnel reflectance matrix. Calculation of Fp follows the specificationgiven above for the Cox-Munk BRDF. The only parameter is the Vegetation Index NDVI, whichvaries from -1 to 1 and is defined as the ratio of the difference to the sum of two radiancemeasurements, one in the visible and one in the infrared.6.3.7. The direct beam correction for BRDFsFor BRDF surfaces, the reflected radiation field is a sum of diffuse and direct (“single-bounce”)components for each Fourier term. One can compute the direct reflected beam with the totalBRDF configured for the solar and line-of-sight directions, rather than use the truncated formsbased on Fourier series expansions. This exact “direct beam (DB) correction” is done before thediffuse field calculation in VLIDORT. Exact upwelling reflection (assuming plane-parallel beamattenuation) to optical depth τ may be written:↑⎡−τ⎤atmos⎡−( τatmos−τ) ⎤IREX( μ,φ,τ ) = I0ρtotal( μ,μ0,φ − φ0)exp⎢⎥ exp⎢⎥ . (6.3.15)⎣ μ0⎦ ⎣ μ ⎦For surface property Jacobians, we require computation of the derivatives of this DB correctionwith respect to the kernel amplitudes and parameters; this follows the discussion in section 2.5.1.For atmospheric profile weighting functions, the solar beam and line-of-sight transmittances inEq. (6.3.15) need to be differentiated with respect to layer variables.115