Atmospheric CO2 – Abstract

It is useful to have a “best-case” expectation of measurement parameters and uncertainties in order to optimize instrumentation methods and limiting parameters. While the proposed method is well established, it is possible to refine the measurement instrumentation to optimize the data integrity when the experimental bounds are established.

The uncertainties related to beam divergence angle and optical path length – essentially the volume of the optical path, the altitude dependency of density of absorption particles within that path, and the mismatch of attenuated laser power to the detector dynamic range contribute significantly more error to the measurement of optical depth than noise in a properly designed electronic system.

If I know the measurement limitations with “best-case” analysis, I can minimize “over-engineering” and perhaps refine a design to optimize results rather than system.

KISS … but not overly so. Several of the geometrical and experiment environment assumptions could be improved.

Dr. McGlone received BSEE and MSEE degrees focusing on analog signal processing techniques for low-level, high-precision sensor interface networks at discrete and integrated circuit levels. He obtained a PhD in electromagnetic geophysics from the Colorado School of Mines with a focus on geophysical instrumentation. He has over 30 years of experience developing geophysical instrument systems including time spent with DoD and DoE ground and space based projects. He joined NASA’s Langley Research Center in 2010.

Abstract

Although there are additional complexities to a field-grade measurement system, a simplified representation of the measurement environment can highlight some sources of uncertainty. The basic concept for this measurement is the absorption of a photon having a specific energy (defined by the photon/laser wavelength) by a CO2 molecule. The loss of this photon from the beam decreases the intensity of the beam.

A continuous beam of uniform density of photons – a constant intensity – is transmitted downward from an aircraft. This beam is reflected from the ground and the return intensity is measured. The decrease in intensity may contain the information desired; however, this process requires two lasers – one “tuned” to be absorbed by CO2, the other tuned to not be absorbed.

The cross-sectional area of the beam diverges at a uniform rate as does the distribution of photons within the beam. Photons are either reflected back to the receiver, scattered out of the optical path, or absorbed within the optical path. To first-order, both beams are subject to identical conditions save one being subject to absorption.

A small diameter laser of the desired wavelength(s) is transmitted normal to the reflecting surface (ground). The beam is reflected back and is gathered by a telescope and directed to a photodetector. It is assumed the transmitter (TX) and receiver (RX) are on the same plane. The ON (absorption) and OFF (non-absorption) wavelengths are assumed to have identical paths and scattering parameters – the only difference is the absorption coefficient.

For this discussion, the beam is assumed normal to the (perfect) reflecting surface. A reasonable altitude for aircraft-based measurements is 10 km (~33,000 ft). A reasonable horizontal velocity is 200 m/s (~450 mph). For reference, light travels the 20 km path distance in 67 $\mu$ s; during this time, the aircraft moves 1.3 cm.

It has been suggested that only 1 of 10 million photons make it to the receiving photodiode*.

The system itself may be considered analogous to one of the earliest of radios – simple amplitude-modulation. The carrier is the laser optical frequency (about 191 THz); absorption and scattering losses modulate the carrier intensity. A photodiode functions as a rectifying broadband detector (as did a chunk of galena in a crystal radio). The photodiode strips off the carrier and the output is a current proportional to carrier envelope (beam intensity).

*conversations with laser scientists at NASA-Langley Research Center

What is actually measured is the number of photons within a particular range of wavelengths (wider than the laser beam wavelengths themselves) striking a photodetector and producing a fundamentally linear current output (number of photons $\propto$ number of free electrons). One of the scaling factors necessary to extract the information desired is the optical path volume in both travel directions – density being number of particles per volume.

The optical path volume is derived from the product of cross-sectional area of the beam as a function of its length. Within this volume are a number of attenuation elements, each with an effective absorption or scattering characteristic.

The dimensionless optical depth calculation is based on the Beer-Lambert Law following the formulation:

$\begin{displaymath}\vartheta\;=\;\int_0^Z\,K\,d z\;=\;N\,\sigma\,\int_0^Z\,d z\;\;\Rightarrow\;\;N_{CO2}[m^{-3}]\,\times\,\sigma[m^2]\,\times\,z[m]\end{displaymath}$

This expression is based on the assumption of uniform particle distribution and constant cross-section. It may be shown that these assumptions decrease in validity for a vertical column of sufficient altitude:

the number of absorption particles in the beam is both altitude and temperature dependent
the effective absorption cross-section of CO₂ is slightly altitude and temperature dependent
the cross-section area of photon flux is dependent on altitude but independent of temperature.

Based on a Standard Atmosphere model*, the particle density decreases exponentially as altitude increases; the photon flux density decreases as the square of the product of beam divergence angle and distance from the laser. Vertical temperature profiles are also dependent on altitude but are not monotonic and not readily modelled.

*wikipedia.org/wiki/U.S._Standard_Atmosphere

The approximation used herein is expressed:

$\begin{displaymath}\vartheta\;=\;\int_0^H\,K\,d z\;\approx\,\sigma\,\int_0^H\,N[z]\,d z\end{displaymath}$

where the effective absorption area is assumed constant but the number of absorption particles is a function of altitude and is contained within the integral.

A 1^st-order sensitivity analysis shows the function sensitivity to the number of particles n to be 1, while the sensitivity to the laser beam divergence angle is 2, and 3 to the distance Z.

The volumetric parameters are a much more significant source of limitation to measurement uncertainty; variations in platform position and orientation, topography, and reflectivity add more inherent error than the desired measurement resolution. Correction algorithms and smoothing techniques further tend to increase uncertainty beyond that desired; sufficient correction tends to a “nominal” (and often “desired”) value.

Initial Development

An understanding of the fundamental process is necessary to develop instrumentation to measure parameters of said process. It is necessary to estimate geophysical parameters in order to define instrumentation compliance range and requirements of system accuracy.

“Optical depth [1]” $\vartheta$ is a measure of electromagnetic (EM) transmission coefficient – “light” being an electromagnetic phenomena. Within a finite media, electromagnetic energy is transmitted, scattered or absorbed depending on the properties of the media it passes through. The measure of optical depth provides a mean of comparing optical transparencies.

Photons may be considered quanta electromagnetic particles with energy proportional to frequency (or the inverse, wavelength).

In the environment of interest, photons of a very specific energy may be absorbed by very specific matching molecules within the optical path. Photons may also be scattered by other less quantifiable elements. The wavelength for this experiment is 1571 nm and the absorbing molecule is carbon dioxide (CO₂). Each photon has energy:

$E_\gamma\;=\;\dfrac{\,h\,c\,}{\lambda}\;\;\Rightarrow\;\;\approx\;1.27\times 10^{-19}\;\textnormal{J}$

where $c$ is speed of light, $h$ is Planck’s Constant, and $\lambda$ is the laser wavelength.

The measure is of the number of photons arriving at the end of the optical path; various signal comparisons are necessary to differentiate between the photons lost to absorption and those lost to scattering. In this experiment, photon fluxes of two different energies are transmitted: one of a specific absorption wavelength, the other slightly off. This assures commonality in path and scattering elements with the only difference between them being the absorption element.

In electrical terms, the measure is path-dependent resistivity using differential techniques. Terminal potentials are known; one path has unknown resistivity, the other has the same unknown resistivity plus an additional resistivity factor. The goal is to determine the magnitude of the added resistivity – CO₂ absorption.

This is an ohmmeter comparable to a Wheatstone Bridge with one unknown optically-resistive element.

The resistances increase exponentially with distance.

It is not enough to know material resistivity to determine resistance; geometry is also a factor. The basic expression is not acceptable in this situation (where neither resistance R nor resistivity $\rho$ are known and cross-section area A is inferred)

$R\;=\;\rho\,\dfrac{z}{\,A\,}$

The expanded expression is still an over-simplistic form which accounts for dependence on the primary geometrical factors of this experiment

$R\;\;\Rightarrow\;\;\dfrac{\rho [z]\,}{\;\pi\;z\;\theta^2}\;}$

where resistivity $\rho$ is a function of distance z and $\theta$ is the beam divergence angle.

[1] A common symbol for optical depth is tau ( $\tau$ ) but as $\tau$ is also commonly used as a symbol for time, optical depth herein will be symbolized as “ $\vartheta$ ”; $\tau$ is reserved for time functions.

That’s all for now.

Next: 2 – Beer-Lambert & Optical Depth

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