Lambert’s Cosine Law and Sidescan Sonar Modeling

When a planar surface area reflects diffusely, the reflected rays have a distribution of directions as opposed to specular reflection which is in a single direction. Ideal diffuse reflection, aka Lambertian reflection, has equal reflected energy at all angles away from the plane. This leads to Lambert's cosine law that says the observed radiance from the area will be proportional to the cosine of the incidence angle. This result has been interpreted in differing ways which we will address here to show that the same reasoning leads to other 'laws' in other situations. The key is that Lambert's law refers to a defined illuminated area of a plane and the energy coming off that area, for an ideal diffuse reflector when irradiated at a particular angle of incidence. It is the result of the geometry that the range of incident angles of light reaching the observer will follow exactly the cosine law. In the case sidescan sonar the area being ensonified also changes with the angle as does the area contributing to the reflected energy in a particular time bin. This changes the cosine law to a cotangent law. The sine in the denominator is due to the changing size of the area being observed at different angles.