<p>I believe your suspicion is correct,otherwise dimensions won't match.</p>

CC BY-SA 4.0
2023-03-14T02:11:28.697Z
Thank you for checking
user8469759
CC BY-SA 4.0
2023-03-13T01:17:01.210Z
2023-03-13T01:17:01.210Z
anonymous5299
<p>I am reading through <a href="https://www.researchgate.net/publication/234800181_Ray_Tracing_Volume_Densities" rel="nofollow noreferrer">Kajiya - Ray Tracing Volume Densities</a> paper. And I've already got stuck into section 2. I wonder if there's a mistake in that equation.</p>
<p>I'll quote the relevant bit</p>
<blockquote>
<p>The quantity to be calculated in a scattering problem is the energy per unit solid angle per unit area
<span class="math-container">$$
dE = I(x,\omega)\sin\vartheta d\omega d\sigma
$$</span>
This quantity is called the <em>intesity</em> of radiation at a point <span class="math-container">$x$</span> in the direction of the solid angle <span class="math-container">$d\omega$</span>. The scattering equation can be derived by considering a differential cylindrical volume <span class="math-container">$dV = d\sigma ds$</span>, where <span class="math-container">$d\sigma$</span> is the cross section of the cylinder and <span class="math-container">$ds$</span> is the length. If we follow a pencil of radiation along the length of the cylinder, we find the difference in intensity between the two ends is given by
<span class="math-container">$$
dI = -absorbed + emitted = -k\rho ds d\sigma d\omega + j \rho ds d\sigma d\omega \;\;\; (2.1)
$$</span>
where <span class="math-container">$\rho$</span> is the density of the matter in the volume element <span class="math-container">$k$</span> is the absorbption coefficient...</p>
</blockquote>
<p>I believe there's a mistake in the absorbed intensity expression, there's a reference quoted at the very beginning of the section where this formulation is taken from (the reference is <a href="https://www.google.co.uk/books/edition/Radiative_Transfer/K7YLRcTP_XUC?hl=en&amp;gbpv=1&amp;dq=chandrasekhar%20radiative%20transfer%20pdf&amp;printsec=frontcover" rel="nofollow noreferrer">Chandrasekhar - Radiative Transfer</a>). I cannot to manage what later the paper defines as scattering equation</p>
<p><span class="math-container">$$
\frac{1}{k\rho} s\cdot\nabla_xI(x,s) - I(x,s) + \frac{1}{4\pi} \int_{\left\lVert s \right\rVert = 1} p(s,\overline{s})I(x,\overline{s}) d\overline{s}
$$</span></p>
<p>I believe in the reference (Chandrasekhar) I've found what was used for Kajiya's derivation, I'll put an image just in case not everyone can see it from the preview:</p>
<p><a href="https://i.sstatic.net/elb9Z.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/elb9Z.png" alt="enter image description here" /></a></p>
<p>In Chandrasekhar's formula 44 there's a <span class="math-container">$I_\nu$</span> factor, which should correspond to <span class="math-container">$I(x,s)$</span> in Kajiya's. And formula (48) in Chandrasekhar is exactly the same as Kajiya's scattering equation, except that the former takes into account in his formulation time and wavelength. But apart from this they're the same.</p>
<p>So I suspect there's a typo in Kajiya's. I think in general absorption is formulate as some form of exponatial decay of the form <span class="math-container">$f'(x) = -\alpha f(x)$</span>, which is used in Chandrasekhar. In summary I think equation (2.1) from Kajiya should be written as</p>
<p><span class="math-container">$$
dI d\sigma d\omega = -k\rho I ds d\sigma d\omega + j \rho ds d\sigma d\omega
$$</span></p>
<p>Can you confirm if my suspicion is correct? Thank you</p>

<mathematics><volumetric-scattering>
Error in equation 2.1. in Ray Tracing Volume Densities
user8469759
# Error in equation 2.1. in Ray Tracing Volume Densities

(Asked by: username_0 on 2022-05-11)

I am reading through [Kajiya - Ray Tracing Volume Densities](https://www.researchgate.net/publication/234800181_Ray_Tracing_Volume_Densities) paper. And I've already got stuck into section 2. I wonder if there's a mistake in that equation.

I'll quote the relevant bit

> The quantity to be calculated in a scattering problem is the energy per unit solid angle per unit area
> $$
> dE = I(x,\omega)\sin\vartheta d\omega d\sigma
> $$
> This quantity is called the *intesity* of radiation at a point $x$ in the direction of the solid angle $d\omega$. The scattering equation can be derived by considering a differential cylindrical volume $dV = d\sigma ds$, where $d\sigma$ is the cross section of the cylinder and $ds$ is the length. If we follow a pencil of radiation along the length of the cylinder, we find the difference in intensity between the two ends is given by
> $$
> dI = -absorbed + emitted = -k\rho ds d\sigma d\omega + j \rho ds d\sigma d\omega \;\;\; (2.1)
> $$
> where $\rho$ is the density of the matter in the volume element $k$ is the absorbption coefficient...

I believe there's a mistake in the absorbed intensity expression, there's a reference quoted at the very beginning of the section where this formulation is taken from (the reference is [Chandrasekhar - Radiative Transfer](https://www.google.co.uk/books/edition/Radiative_Transfer/K7YLRcTP_XUC?hl=en&gbpv=1&dq=chandrasekhar%20radiative%20transfer%20pdf&printsec=frontcover)). I cannot to manage what later the paper defines as scattering equation

$$
\frac{1}{k\rho} s\cdot\nabla\_xI(x,s) - I(x,s) + \frac{1}{4\pi} \int\_{\left\lVert s \right\rVert = 1} p(s,\overline{s})I(x,\overline{s}) d\overline{s}
$$

I believe in the reference (Chandrasekhar) I've found what was used for Kajiya's derivation, I'll put an image just in case not everyone can see it from the preview:

[![enter image description here](https://i.sstatic.net/elb9Z.png)](https://i.sstatic.net/elb9Z.png)

In Chandrasekhar's formula 44 there's a $I\_\nu$ factor, which should correspond to $I(x,s)$ in Kajiya's. And formula (48) in Chandrasekhar is exactly the same as Kajiya's scattering equation, except that the former takes into account in his formulation time and wavelength. But apart from this they're the same.

So I suspect there's a typo in Kajiya's. I think in general absorption is formulate as some form of exponatial decay of the form $f'(x) = -\alpha f(x)$, which is used in Chandrasekhar. In summary I think equation (2.1) from Kajiya should be written as

$$
dI d\sigma d\omega = -k\rho I ds d\sigma d\omega + j \rho ds d\sigma d\omega
$$

Can you confirm if my suspicion is correct? Thank you

**Tags:** mathematics, volumetric-scattering


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## 1 answers

### Answer by username_1 on 2023-03-13. Score: 1 | Accepted answer

I believe your suspicion is correct,otherwise dimensions won't match.

> **Comments:**
> **username_0**: Thank you for checking  

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