Method To Measure Atmospheric Molecular Absorption

One topic I had been interested in for some time is the problem of measuring atmospheric constituent gases from aircraft using molecular absorption methods. I was lucky enough to be employed at NASA where some of this research was based

Various gaseous constituent molecules tend to absorb photons of very specific wavelengths – so the principle idea of measurement is that by measuring a change of intensity of a specific optical wavelength, the absorbing gas can be identified.

Lasers are useful for transmitting photons of specific wavelengths …

There are many complexities that may be added to the problem, but let’s start by considering simplicity. Since everybody loves CO2, I’m going to use the measurement of CO2 concentration as an example. CO2 being considered ubiquitous in the lower atmosphere (troposphere) and the Standard Atmospheric Model being essentially linear in this region, the use of these will be considered baseline assumptions. Furthermore, it will also be assumed that the Beer-Lambert Law is an adequate representation of the atmospheric absorption characteristics over relatively long distances.

Basic Description

The absorbance factor A is equal to the log of the ratio of received intensity to transmitted intensity:

$\begin{displaymath}\mathcal{A}\,=\,\mathit{ln}\frac{\Phi_{RX}}{\,\Phi_{TX}\,}\end{displaymath}$

It can be shown that A is equal to the product of molar density $\delta$ , cross-sectional area $\sigma$ of the absorbing molecule, and the optical path length z.

$\begin{displaymath}\mathcal{A}\,=\,\delta\,\sigma\,z\end{displaymath}$

where absorbance $\mathcal{A}$ is dimensionless.

The measurement goal is to determine molar density $\delta$ .

Re-arranging:

$\begin{displaymath}\mathcal{A}\,=\,\mathit{ln}\frac{\Phi_{RX}}{\,\Phi_{TX}\,}\;\rightarrow\;\frac{\Phi_{RX}}{\,\Phi_{TX}\,}\,=\,\mathit{e}^\mathcal{A}\;\rightarrow\;\Phi_{RX}\,=\,\Phi_{TX}\,\mathit{e}^\mathcal{A}\end{displaymath}$

where $\Phi_{TX}$ is the transmitted laser intensity – a known quantity – and $\Phi_{RX}$ is the measured quantity of the reflected laser intensity. The value of $\mathcal{A}$ is determined from $\Phi_{RX}$ … where both molecular cross -section of the species of interest and optical path length are (assumed) known quantities.

$\begin{displaymath}\Phi_{RX}\,=\,\Phi_{TX}\,\mathit{e}^\mathcal{A}\;\equiv\;\Phi_{RX}\,=\,\Phi_{TX}\,\mathit{e}^{\left(\delta\,\sigma\,z\right)}\end{displaymath}$

Sensitivity

The sensitivity of a function y to a change in parameter x is defined:

$\begin{displaymath}S_x^y\;=\;\frac{\,x\,}{y}\,\frac{\,\partial y\,}{\partial x}\end{displaymath}$

Applying this to the absorption function for sensitivity to optical path length z

$\begin{displaymath}S_x^y\;=\;\frac{\,x\,}{y}\,\frac{\,\partial y\,}{\partial x}\;\Rightarrow\;\frac{z}{\,\Phi_{RX}\,\mathit{e}^{\left(\delta\,\sigma\,z\right)}}\,\frac{\,\partial\left(\Phi{RX}\,\mathit{e}^{\left(\delta\,\sigma\,z\right)}\,}{\partial z}\;=\;\delta\,\delta\,z\end{displaymath}$

Similarly for molecular density $\delta$

$\begin{displaymath}S_x^y\;=\;\frac{\,x\,}{y}\,\frac{\,\partial y\,}{\partial x}\;\Rightarrow\;\frac{\delta}{\,\Phi_{RX}\,\mathit{e}^{\left(\delta\,\sigma\,z\right)}}\,\frac{\,\partial\left(\Phi{RX}\,\mathit{e}^{\left(\delta\,\sigma\,z\right)}\,}{\partial \delta}\;=\;\delta\,\delta\,z\end{displaymath}$

The absorbance function is equally sensitive to optical path length and molecular density: i.e., a 1% change in optical length is indistinguishable from a 1% change in molecular density. This can be a measurement issue …

This is a major consideration when it comes time to declare instrument capability and measurement error. Consider an aircraft at 10 km altitude (~ 32,000 ft). The optical path length is 20 km. A 1% shift in altitude is a 100 m change in altitude. NASA suggests that one of their primary research aircraft has a GPS-based vertical uncertainty of 15 m.

The assurance of an acceptable measurement accuracy of one term of a two-term product usually requires the accuracy of the second term to be roughly 10x better than the desired measurement accuracy of the first. Consider the total error if both variables are accurate to 1%.

$\begin{displaymath}\varepsilon\;=\;\sqrt{\,\left(\frac{\,0.01\,}{1}\,\right)^2\,+\,\left(\frac{\,0.01\,}{1}\,\right)^2\,+\,2\,\left(\,\frac{\,0.01\times0.01\,}{1\times 1}}\right)\,}\;=\;0.02\,\textnormal{or}\; 2\%\end{displaymath}$

If the parameter of the secondary interest is known to 0.1% (10x better), then:

$\begin{displaymath}\varepsilon\;=\;\sqrt{\,\left(\frac{\,0.01\,}{1}\,\right)^2\,+\,\left(\frac{\,0.001\,}{1}\,\right)^2\,+\,2\,\left(\,\frac{\,0.01\times0.001\,}{1\times 1}}\right)\,}\;=\;0.011\,\textnormal{or}\; 1.1\%\end{displaymath}$

This defines the limit of measurement error: if it is desired that the total measurement error $\varepsilon$ be a certain value, then the combination of parameter errors must be equal to or less than that desired error. Because the desired measurement is of molecular density, it is the optical path length that needs to be known to a high degree of accuracy.

Measurement Methodology

Simply measuring the intensity of a reflected beam will not provide the desired information – it is necessary to have knowledge of the nominal decrease in beam intensity without absorption; the optical path is not ideal and the reflection coefficient of the reflecting surface is a major contributor to intensity attenuation. One method of accomplishing the desired measurement is by using a version of correlated double sampling (CDS) in which the unknown value is compared to a known value – a difference measurement.

This can be accomplished by the transmission of a laser wavelength which is very close to the absorption wavelength – but far enough off the absorption wavelength to be immune to absorption attenuation. The basis for the acceptance of this technique is that both beams are transmitted through the same optical path and therefore subject to the same parasitic attenuation factors. Under the premise of this basic assumption, there are multiple methods in which this may be accomplished.

Difference Measurement

Optical depth* is a term often used for this measurement. Essentially equivalent to parameter A, it may be defined as:

$\begin{displaymath}\mathcal{OD}\,=\,N(cm^3)\,\times\,\sigma(cm^2)\,\times\,dz(cm)\end{displaymath}$

where N is the volumetric concentration of the material of interest (CO2 in this example), $\sigma$ is the cross-sectional are of the absorption molecule (considered constant to 1^st-degree), and optical path length z. Since the function is assumed linear, the units may be expanded to meters with no loss of generality.

There will be two laser channels: one “on” the absorption wavelength, the other “off” the absorption wavelength. Both are subject to the Beer-Lambert effect, these will be defined as:

$\begin{displaymath}\mathcal{A}_{on}\,=\,\mathit{ln}\frac{\Phi_{RX,on}}{\,\Phi_{TX,on}\,}\qquad \mathcal{A}_{off}\,=\,\mathit{ln}\frac{\Phi_{RX,off}}{\,\Phi_{TX,off}\,}\end{displaymath}$

It is assumed that the transmitted intensities are known.

$\begin{displaymath}\frac{\frac{\Phi_{RX,on}}{\,\Phi_{TX,on}\,}}{\,\frac{\Phi_{RX,off}}{\,\Phi_{TX,off}\,}\,}\;=\;\frac{\Phi_{RX,on}}{\,\Phi_{RX,off}\,}\,\frac{\Phi_{TX,off}}{\,\Phi_{TX,on}\,}\;=\;\frac{\Phi_{TX,on}\,\mathit{e}^{\mathcal{A}on}}{\,\Phi_{TX,off}\,\mathit{e}^{\mathcal{A}off}\,}\,\frac{\Phi_{TX,off}}{\,\Phi_{TX,on}\,}\;=\;\mathit{e}^{\left(\mathcal{A}on-\mathcal{A}off\right)}\end{displaymath}$

Since:

$\begin{displaymath}\mathcal{A}on \,=\,\left(\,\delta s\,\sigma s\,+\,\delta a \, \sigma a\right)\, z \qquad \qquad \mathcal{A}off \,=\,\left(\,\delta s\,\sigma s\,\right)\, z\end{displaymath}$

where $\delta s$ and $\sigma s$ are the effective non-absorption attenuation factors.

Substituting terms

$\begin{displaymath}\frac{\frac{\Phi_{RX,on}}{\,\Phi_{TX,on}\,}}{\,\frac{\Phi_{RX,off}}{\,\Phi_{TX,off}\,}\,}\;=\;\mathit{e}^{\left(\mathcal{A}on-\mathcal{A}off\right)}\;=\;\mathit{e}^\left(\delta a\,\sigma a\,z\right)\;\Rightarrow\;\mathit{ln}\left[\mathit{e}^\left(\delta a\,\sigma a\,z\right)\right]\;=\;\delta a\,\sigma a\,z\end{displaymath}$

for which only $\delta a$ is unknown.

Now we only have to implement such a function in some sort of electronic implementation.

Easy! Digitize, store, and numerical processing! Oh, ye of little faith …

Time & Space

First, consider the factor of time. If the aircraft maintains a constant altitude of 10 km, the optical path is a constant 20 km (this assumes the aircraft accurately follows the terrain – which is not likely).

The speed of light can be approximated as 300e6 m/s with less than 0.1% error.

In terms of time, the optical path length is 2 $\times$ 33.3 μs (the path should be separated into “before” and “after” reflection as the beam characteristics are different for these segments of the path). In terms of electronic processing, these are moderate speeds – a not-unreasonable 100 MHz, 50% duty-cycle sample period is 5 ns.

One method of measurement could be to “gate” the laser and sample the gate period. For a 70 μs gate, there would be 14,000 samples. This seems attractive but generates a tremendous amount of data over a typical flight length of 4 hours. If continuous sampling is desired, this would generate 5.24 Tbytes of data during a 4-hour flight … per sampled channel.

Such quantities of data are not really necessary. Consider the measurement in terms of spatial content. Assume an aircraft velocity of 200 m/s (447 mph)

One question would be related to spatial requirements: is data necessary every meter?, 10 meters?, 100 meters?

How much spatial resolution of CO2 concentration is useful for an aircraft-based measurement??

Let’s say 100 m is a useful sample period. Let’s further consider that it may be desirable to average 10 distinct samples for each 100 m data point. This suggests sub-data at 10 m intervals or a sample point every 5 ms at a velocity of 200 m/s.

Integration & Averaging

The conversion of photons to electrons naturally lends itself to integration. It is desired to obtain the “average” value of photons detected over some period of time. Furthermore, one may consider the operation of the quantization process – a signal magnitude is “held” during the conversion process; the implementation of “hold” often requires an accumulation of charge on a capacitor One definition of capacitance is the ratio of electric charge to electric potential.

$\begin{displaymath}C\,=\,\frac{\,Q\,}{V}\;\Rightarrow\;Q\,=\,C\,V\end{displaymath}$

Electrical current is the movement of electrons:

$\begin{displaymath}I\,=\,\frac{\,dq\,}{dt}\end{displaymath}$

so that

$\begin{displaymath}I\,=\,\frac{\,dq\,}{dt}\;=\;C\,\frac{\,dV\,}{dt}\;\Rightarrow\;dq\,=\,C\, dV\end{displaymath}$

So the photons interact with the photodetector to create electrons (which are quanta elements of charge). A change of photon intensity causes a corresponding change of charge intensity. This charge accumulates; a change of charge causes a change in electric potential which manifests itself as a measurable (hopefully) voltage:

$\begin{displaymath}\int_0^T\,\frac{\,dq\,}{dt}\,dt\,=\,C\,\int_0^T\,\frac{\,dV\,}{dt}\,dt\;\Rightarrow\;Q\,=\,C\,V\end{displaymath}$

What we do is modify the integration period to fit the desired spatial integration period desired.

Averaging can be accomplished discretely:

$\begin{displaymath}AVG_D\;=\;\frac{1}{\,N\,}\,\sum_{n=1}^N\,x^n\end{displaymath}$

or continuously:

$\begin{displaymath}AVG_C\;=\;\frac{1}{\,T\,}\,\int_0^T\,x(t)\,dt\end{displaymath}$

The discrete method requires N samples to be stored and further processed by summation and division; the continuous method produces an output already “processed” … with a simple circuit without the addition of software beyond integration time control.

*Optical depth is usually designated $\tau$ , however, I use $\tau$ to define some aspect of time. There optical depth will be designated “OD”

There is a longer, more in-depth article – “CO2 – Considerations For Atmospheric Measurement” posted here.

Leave a Comment Cancel Reply