Parameter error propagation in BRDF derived by fitting multiple angular observations at single sun position

In most recently developed linear kernel-driven BRDF (bidirectional reflectance) models, there are usually 3 unknowns for each band. Usually a least square (LS) approach is employed for inversion. Assuming the observations are well sampled over the viewing hemisphere (viewing zenith angle θ_υ, azimuthal difference φ with the solar zenith angle θ_i) for a single θ_i, as in most cases of space-borne multiangular observations such as POLDER and MISR, the LS solution can be obtained for the three unknowns. It was once hoped that if the kernel-driven model has sound physical meaning, the three parameters estimated from such good 2-D sampling can be used over the whole 3-D (θ_i, θ_υ, φ) bidirection space (3DBS for short). However, inversion of 395 BRDF datasets acquired by POLDER of CNES (France) shows that when we apply the inversion results over the whole 3-D space, for example, at a far different θ_i, the estimation errors in parameters will propagate differently and thus yield different pattern of prediction errors, independent of the soundness of the BRDF model physics. Our analysis concludes that general knowledge of BRDF shapes of the land surface has to be applied to constrain the inversion of single (or narrow-range) θ_i multiangular observations