There exist some basic models for the angle dependence of sigma0. In R.E. Clapp, 1946, “A theoretical and experimental study of radar ground return” three such models are presented:
[1] sigma0(theta) = constant * cos(theta)2 called: “Lambert’s law”
[2] sigma0(theta) = constant
[3] sigma0(theta) = constant * cos(theta)
[3] is actually more complicated since it can also include multiple reflections from deeper layers of the surface. If however one only considers direct reflections the model takes the form as shown above.
In Ulaby, Moore, Fung, 1982, “Microwave Remote Sensing, Active and Passive” vol. II the authors also discuss these models of Clapp.
With models [1] and [3] one cosine(theta) term accounts for the decrease in incident power per unit surface area when the radar measures the ground return under angle theta. With [1] a seccond cosine(theta) term is added in accordance with Lambert’s law: a radiating surface whose angle-dependent emission is according to I = I0 * cos(theta) [Wm-2].
The well-known integral form of the radar equation applied to surface returns is (see for example Ulaby1982):
Prx = Ptx * [ lambda2 / (4 pi)3 ] * integral[ G2 / R4 * sigma0(theta) , dA ]
What I don’t understand is why there is not a cosine term in this equation by default? So
Prx = Ptx * [ lambda2 / (4 pi)3 ] * integral[ G2 / R4 * sigma0(theta) * cos(theta) , dA ]
Because the way I see it: regardless of the scattering properties of any surface the incident power per unit surface area must be rescaled according to cos(theta).
@ Jianping Wang: Thank you for your answer, I appreciate it.
I am familiar with the concept of the radar cross section (RCS). In terms of general scattering theory the RCS is actually the differential scattering cross section. Like any scattering cross section also the RCS can have a value [m^2] that can be several magnitudes larger or smaller than its physical cross section.
You mention in with sigma0 the aspect angle of observation is already implicitly considered. That got me thinking about the 3 models of Clapp mentioned above. With [1] sigma0(theta) = K1 cos(theta)^2 and [3] sigma0(theta) = K3 cos(theta) the incident intensity is projected to a unit surface area, hence the cos(theta)^1 term. With [2] sigma0(theta) = K2, a constant. The idea of the model is that the surface is a single layer of metal spheres, that are apart from each other far enough that only for very high angles theta they start to be in each others line of view with respect to the radar. As a result the projection of incident intensity to the surface is constant, no matter what angle, since you see the same amount of sphere area all the time.
What these 3 cases show is that Clapp made the cosine projection explicit, depending on the model, [1] and [3] have it, [2] does not. Is this what you mean by your statement that sigma0 implicitly considers the effects of aspect angle of observation?
From the above it then makes sense that the radar equation does not have the cosine term by default. Whether there is a cosine therm in the integral depends solely on the model of sigma0.
I am curious whether you agree with my story.
Hi Jan Hofste , I'm also really interested in this question. I don't have access to the original Clapp text but have of course seen it referenced in the literature. Have you thought any more about this since you first asked the question? From your response above it sounds like you had reasoned that sigma0 does not have the cosine term in the radar equation because model [2] is assumed--is that what you were thinking? Thanks!
@ C.Chew.
The way I see it is that the cosine term, that accounts for the incident intensity being projected onto a unit surface area, is only present if it is part of the model of sigma0(theta) that you use.
So I don't think that when you see the radar equation as:
Prx = Ptx * [ lambda2 / (4 pi)3 ] * integral[ G2 / R4 * sigma0(theta) , dA ]
that then model [2] of Clapp is used. The radar equation with Clapp's model [2], I think, will look like:
Prx = Ptx * [ lambda2 / (4 pi)3 ] * integral[ G2 / R4 * constant , dA ]
The form
Prx = Ptx * [ lambda2 / (4 pi)3 ] * integral[ G2 / R4 * sigma0(theta) , dA ]
is just the generic notation of the radar equation, implying that some angle-dependent model for sigma0 is used. In the case of Clap [2] this is simply a constant.
I can send you a PDF of Clapp's report if you are interested.
@ Jan Hofste: Thanks for asking the question above - it's been something I've been recently trying to figure out. I've also been looking for a PDF copy of Clapp's 1946 report. In your last post you offered to send it to @ C.Chew, could I ask you to send it to me also? I've searched extensively for the report, but have not be able to source a copy. Thanks in Advance, Daithí