<p>The factor <span class="math-container">$e^{-\tau_d (1/\cos \theta_i + 1/\cos \theta_r)}$</span> represents the attenuation of backscattered light due to the path it takes through the layer.</p>
<p>Imagine for a moment a layer of thickness 1. A ray passing through this layer from top to bottom at angle <span class="math-container">$\theta$</span> will then have a length of <span class="math-container">$1/\cos \theta$</span>, which you can see by drawing a right triangle. This applies to both the incident and reflected rays, so the total path length of the ray within this layer is <span class="math-container">$1/\cos \theta_i + 1/\cos\theta_r$</span>.</p>
<p>Multiply this by the optical thickness of the layer, <span class="math-container">$\tau_d$</span>, and you have the optical thickness of the whole ray path; thus <span class="math-container">$e^{-\tau_d (1/\cos \theta_i + 1/\cos \theta_r)}$</span> is the attenuation factor along that path.</p>
<p>The other cosine factors out front and the <span class="math-container">$1 -$</span> in front of the exponential term, I'm guessing, probably arise from doing an integral over the depth of the layer, to account for scattering events at any depth in the layer. This often shows up when integrating exponentials:
<span class="math-container">$$
\int_0^{\tau_d} e^{-\tau \cdot \text{stuff}}\,d\tau
= \frac{1}{\text{stuff}} \bigl(1 - e^{-\tau_d \cdot \text{stuff}} \bigr)
$$</span>
as long as the <span class="math-container">$\text{stuff}$</span> is independent of depth.</p>

CC BY-SA 4.0
2021-12-24T04:51:59.087Z
2021-12-24T04:51:59.087Z
Nathan Reed
<p>Recently, I am doing some research about subsurface scattering.  i am a little confused about the <strong>backscattered radiance</strong> mentioned in this paper <a href="https://cseweb.ucsd.edu/%7Eravir/6998/papers/p165-hanrahan.pdf" rel="nofollow noreferrer">Reflection from Layered Surfaces due to Subsurface Scattering</a>. I understand the light transfer equation, but i just cannot figure out why it has to be that form.</p>
<p><a href="https://i.sstatic.net/vg9Wo.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/vg9Wo.png" alt="enter image description here" /></a></p>
<p>I don't know Why does it has to be <a href="https://i.sstatic.net/nO1zv.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/nO1zv.png" alt="enter image description here" /></a>,there is a picture which can describe the radiance clearly:
<a href="https://i.sstatic.net/68Gk7.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/68Gk7.png" alt="enter image description here" /></a></p>
<p>Does anyone who have read that paper? I need some help</p>

<lighting><brdf><subsurface-scattering>
[Subsurface Scattering]Reflection from Layered Surfaces due to Subsurface Scattering
danding
# [Subsurface Scattering]Reflection from Layered Surfaces due to Subsurface Scattering

(Asked by: username_0 on 2021-12-23)

Recently, I am doing some research about subsurface scattering. i am a little confused about the **backscattered radiance** mentioned in this paper [Reflection from Layered Surfaces due to Subsurface Scattering](https://cseweb.ucsd.edu/%7Eravir/6998/papers/p165-hanrahan.pdf). I understand the light transfer equation, but i just cannot figure out why it has to be that form.

[![enter image description here](https://i.sstatic.net/vg9Wo.png)](https://i.sstatic.net/vg9Wo.png)

I don't know Why does it has to be [![enter image description here](https://i.sstatic.net/nO1zv.png)](https://i.sstatic.net/nO1zv.png),there is a picture which can describe the radiance clearly:
[![enter image description here](https://i.sstatic.net/68Gk7.png)](https://i.sstatic.net/68Gk7.png)

Does anyone who have read that paper? I need some help

**Tags:** lighting, brdf, subsurface-scattering


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## 1 answers

### Answer by username_1 on 2021-12-24. Score: 0 | Accepted answer

The factor $e^{-\tau\_d (1/\cos \theta\_i + 1/\cos \theta\_r)}$ represents the attenuation of backscattered light due to the path it takes through the layer.

Imagine for a moment a layer of thickness 1. A ray passing through this layer from top to bottom at angle $\theta$ will then have a length of $1/\cos \theta$, which you can see by drawing a right triangle. This applies to both the incident and reflected rays, so the total path length of the ray within this layer is $1/\cos \theta\_i + 1/\cos\theta\_r$.

Multiply this by the optical thickness of the layer, $\tau\_d$, and you have the optical thickness of the whole ray path; thus $e^{-\tau\_d (1/\cos \theta\_i + 1/\cos \theta\_r)}$ is the attenuation factor along that path.

The other cosine factors out front and the $1 -$ in front of the exponential term, I'm guessing, probably arise from doing an integral over the depth of the layer, to account for scattering events at any depth in the layer. This often shows up when integrating exponentials:
$$
\int\_0^{\tau\_d} e^{-\tau \cdot \text{stuff}}\,d\tau
= \frac{1}{\text{stuff}} \bigl(1 - e^{-\tau\_d \cdot \text{stuff}} \bigr)
$$
as long as the $\text{stuff}$ is independent of depth.


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