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math question about "extinction coefficient"/optical depth

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I am working to implement this paper on cloud rendering: http://www.markmark.net/PDFs/RTClouds_HarrisEG2001.pdf Iv been stuck for a while on Eq. (1) :P in eq. (1), the integrals in the exponential's are "τ(t)dt" directly quoted from the paper: "τ(t) is the extinction coefficient (in units of 1 / length) of the cloud at depth t..." Iv been googling like crazy for "extinction coefficient" but everywhere I look seems to different formulas, terms, ect... (sometimes I even see it as "optical depth") So my question is, can anyone write out the actual formula for this "τ(t)"? How does this relate to optical depth (or is it optical depth) and/or what does this value actually represent physically? None of these people that write papers can ever use plain english / clear formulas lol thanks a bunch, chris [Edited by - coderchris on July 9, 2008 3:48:16 PM]

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Take a look here and here. Start with the Wolfram equations. You have the formula dτ = -Kdz, where 'K' is your extinction coefficient. So instead of a constant 'K', replace it with the function τ(t), which gives you dτ = -τ(t)dt ('z' and 't' are the same thing here). Integrate both sides with respect to 't' and you get -τ = ∫ab[τ(t)dt]. Now moving on to the Wikipedia equations, you get I = (I0)e = (I0)eab[τ(t)dt]. τ(t) itself is user-defined and depends on the medium you're modeling. In the simplest case, τ(t) would just be a constant like 'K', however it depends entirely on the medium you're modeling.

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Thanks for the reply, but im still pretty confused

So from what I understand, the T(t) that the author refers to in the paper is some arbitrary function. How is one supposed to figure out what the function / constant is? Is there any reference that describes what a good optical depth function is for clouds?

The optical depth function T(t) itself does not contain an integral though, right?


Quote:

You have the formula dτ = -Kdz, where 'K' is your extinction coefficient. So instead of a constant 'K', replace it with the function τ(t),


Why replace K with T(t)?

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Quote:
Original post by coderchris
So from what I understand, the T(t) that the author refers to in the paper is some arbitrary function. How is one supposed to figure out what the function / constant is? Is there any reference that describes what a good optical depth function is for clouds?

Digging a little deeper on Wolfram reveals the equation K = σn, where σ is the cross-sectional area and n is the number density. Number density is like mass density (kg.m-3), except it's the number of objects per unit volume, as opposed to mass per unit volume. Hence it has the units (m-3) since the number of objects (in this case atoms/molecules) is unit-less. In most cases you'd be interested in, the number density is P/(kT) where 'P' is the pressure of the gas, 'T' is the temperature, and 'k' is Boltzmann's constant. IIRC the cross-sectional area is calculated using the Bohr radius of a gas atom, but don't quote me on that (I'll look it up and let you know).

Quote:
Why replace K with T(t)?

It's a more general form that allows the coefficient to vary along the path traveled. For instance, in the equation K = σn we might want temperature to vary along the path. In that case you would need K(t) = σn(t), where 'τ(t)' is conventionally used instead of 'K(t)'.

Quote:
The optical depth function T(t) itself does not contain an integral though, right?

Maybe, but it's very unlikely you wouldn't be able to simplify it away.

In your case, τ(t) will likely be just a constant, as determined by cross-sectional area and number density, both of which you can consider constant. I'll dig up my old astrophysics book tomorrow and go over the chapter that covers optical depth, since it provides some very detailed explanations (in the context of electron drift and the atmosphere on the surface of stars, but still relevant).

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Hmm, ok... starting to make a little more sense

I remembered seeing "optical depth" / "extinction coefficient" in Nishita's famous atmosphere rendering paper. So I looked at that real quick to see if there was any correlation between the wolfram formulas and Nishita's formulas.

His paper is here by the way, in case you want to look at it:
http://portal.acm.org/citation.cfm?id=166140

From Nishita's paper, optical depth T(S) = B * ∫ p(s) ds
where B is what he calls "extinction ratio" and p(s) is "density ratio"

Now, p(s) I think understand. It is simply the density of particles per unit volume, or the number density (as you put it). In my case it makes sense that this should be just a constant. In Nishita's paper, p(s) = exp(-h/H0) which also makes sense because the density of particles should decrease as you go higher in the sky.

If Im interpreting it correctly, p(s) corresponds to the p in the wolfram equation dT = -kp dz

integrating both sides of the wolfram we get
T(s) = -k * ∫ p(s) ds

So the only thing left I need to figure out is what the -k is here. Wolfram defines it as "opacity". Nishita defines it as "extinction ratio"
Since T(S) = B * ∫ p(s) ds (from Nishita)
and T(s) = -k * ∫ p(s) ds (from wolfram)
then that implies that B = -k

where Nishita defines B as 4*pi*K/alpha^4
where K = 2*pi^2 * (n^2 - 1)^2 / 3*Ns
where n = index of refraction of medium, alpha = wavelength of light,
Ns = molecular number density

So, im not quite understanding how -k = B makes any sense, because wolfram sais that k is opacity, and Nishita defines B as that crazy formula...

Does any of this make sense?

thanks again, I appreciate the help

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