Starting from the Rayleigh-Rice perturbation theory, this paper derives the empirical Harvey scatter model and ABg scatter model applied extensively in optical analysis software packages and verifies the shift-invariant behavior of the scattered radiance in direction cosine space. Using data obtained from multi-wavelength laser scatterometer on carbon nanotube black coating and pineblack coating, we establish the polynomial model based on the sine of the scattering angle plus the sine of the specular reflection angle, i.e., sin θs+sin θ0 and the dual-Harvey model based on sin θs-sin θ0 , respectively. The models are in good accordance with the experimental data and further extend the valid range of empirical models. © 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement.