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The attenuation of a wave through a medium can be modeled by the Beer-Lambert Law using an attenuation coefficient. If $I$ is the intensity, and $I_r$ is a reference intensity, then what is the physical difference between modelling attenuation using the two functions below?

I am looking for a conceptual understanding of the difference between $I_1$ and $I_2$. What does the integral in $\beta_1(\tau,\omega)$ imply?

In the equations below, $\tau$ is the time, $\omega$ is angular frequency, and $Q(\tau)$ is a real-valued function of time $\tau$.

I have seen both $\beta_1$ and $\beta_2$ functions used in the context of the attenuation of seismic waves through layered media, and I am wondering if one model is perhaps better than the other.

${I_1} = {I_r}{\beta _1}(\tau ,\omega )$

${I_2} = {I_r}{\beta _2}(\tau ,\omega )$

${\beta _1}(\tau ,\omega ) = \exp \left[- {\int\limits_0^\tau {\frac{\omega }{{2Q(\tau ')}}d\tau '} } \right]$

${\beta _2}(\tau ,\omega ) = \exp \left[- {\frac{{\omega \tau }}{{2Q(\tau )}}} \right]$

A reference for these two equations is Y. Wang, Seismic Inverse Q Filtering (link). See for example, pg. 121.

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It seems to me that $\beta_2$ is simply $\beta_1$ with the assumption that $Q(\tau)$ is approximately constant over the range $[0,\tau]$. Whether one is better than the other, I can't say. Could you give some references for your expressions for the $\beta$s? Neither the Wikipedia page on the Beer–Lambert Law nor the attentuation coefficient mention that integral. – Flavin Feb 25 '13 at 18:45
@Flavin: Thanks Flavin; I think you are right. I've now included a reference in the main body of my post. – Nicholas Kinar Feb 26 '13 at 1:09

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