Theory of operation
Relating absorption to concentration
The scaling law of Jaimeson et. al., (1963) shows the effect of pressure on infrared absorption. If the amount of absorber of some gas u_{i} (mol m^{-2}) and absorption in a band are related by some function h_{i}(), then
The subscript i denotes a particular (i^{th}) gas. Pressure is denoted as P_{ei} because it is the equivalent pressure for the i^{th} gas. Equivalent pressure is potentially different from total pressure P if there are gases present other than i that affect how the i^{th} gas absorbs radiation.
We rewrite this in terms of number density (mol m^{-3}) by introducing a path length λ, and noting that u_{i} = ρ_{i}λ. Substituting this into equation 11‑1, and solving for the number density ρ_{i} of gas i yield
We rewrite equation 11‑2 as
by combining λ and the inverse h() functions into a new function f_{i}(). The calibration function f_{i}() is generated by measuring a range of known densities ρ_{i} and fitting a curve to ρ_{i}/P_{ei} plotted against αi/P_{ei}. Since gas standards are not available in “known densities”, the ρ_{i} values are computed from known concentrations m_{i} (moles of gas per mole of air) using the ideal gas law
11‑4
Measuring absorptance
Given a source with radiant power F, and a detector some distance away, in the absence of reflection, absorptance by gas i can be determined from
11‑5
where is transmittance through gas i, is transmitted radiant power in the absorption band with some concentration of gas i present, and is the transmitted radiant power in the absorption band with zero concentration of i present. The instrument approximates absorptance by
where A_{i} is the power received from the source in an absorbing wavelength for gas i, and A_{io} is the power received from the source in a reference wavelength that does not absorb gas i. The instrument measures A_{i} and A_{io} alternately 150 times per second.
If we combine equations 11‑6 and 11‑3, we can write the full equation for computing molar density from absorptance.
Note the zeroing term z_{i} and the span adjustment term S_{i} in equation 11‑7. The span adjustment term is a linear function of absorptance (see What actually happens):
Cross sensitivity
Because the instrument uses one detector for measuring A_{c}, A_{co}, A_{w}, and A_{wo}, (the absorbed and non-absorbed power for CO_{2} and H_{2}O, respectively), there is a slight cross-sensitivity between gases due to imperfections in the detector's frequency (time) response. This varies from detector to detector, but is measured during calibration, and is corrected in software. Equation 11‑6 is written as
where X^{ji} is the cross sensitivity response of gas j on gas i (determined during calibration), and A_{j} and A_{jo} are the absorbed and non-absorbed power for gas j. Equation 11‑7 becomes
Zero drift
Even though the detector and filters are temperature controlled in the LI-7500DS, the detector is subject to slight temperature drift as ambient temperature changes. This error is directly related to the detector cooler control voltage, which is measured, and thus provides a mechanism for a software "fine tuning".
The zero term z_{i} is computed from
11‑11
where V_{d} is the cooler voltage, Z_{i} is the slope of the relationship between V_{d} and z_{i} (determined during calibration), and Z_{io} is the zero factor determined when setting the zero.
Equation summary
H_{2}O
In the atmosphere, the absorption of radiation by water vapor is not significantly influenced by any other gas, so the effective pressure for water vapor, P_{ew}, is simply the total pressure P.
11‑12
H_{2}O absorptance, a_{w}, is
where b_{1}, b_{2}, and b_{3} are constants (CO_{2} SD1, SD2, and SD3 on the calibration sheet) and V_{c} is cooler voltage. Uncorrected absorptance, a*_{w}, is given by
11‑14 ,
where A_{w} and A_{wo} are the raw signals for the water vapor absorption and reference bands, X_{cw} is the cross sensitivity coefficient for CO_{2} on water vapor (H_{2}O XS on the calibration sheet), A_{c} and A_{co} are the raw signals for the CO_{2} absorption and reference bands, Z_{wo} is the zeroing coefficient (H2O Zero on the calibration sheet), Z_{w} is the zero drift coefficient (H_{2}O Z on the calibration sheet), and V_{c} is the cooler voltage.
Mole density of H_{2}O, ρ_{w}, is given by
11‑15
The coefficients for the 3rd order polynomial f_{w}() are given on the calibration sheet. The polynomial has the form Ax + Bx^{2} + Cx^{3}, where x = a_{w}S_{w}/P. S_{w} is span for H_{2}O.
CO_{2}
The absorption of radiation by CO_{2} molecules is influenced by several other gases, including O_{2} and H_{2}O. Since the concentration of H_{2}O is most variable, it must be accounted for in the equivalent pressure of P_{e}. A method of doing this (LI-COR Application Note #116) is
11‑16
P is pressure, a_{w} is the band broadening coefficient, and m_{w} is the mole fraction of water vapor. a_{w} has been determined to be 1.15 for the LI-7500DS.
CO_{2} absorptance, a_{c}, is given by
where b_{1}, b_{2}, and b_{3} are constants (H_{2}O SD1, SD2, and SD3 on the calibration sheet) and V_{c} is cooler voltage. Uncorrected absorptance, a*_{c}, is given by
11‑18 ,
where A_{c} and A_{co} are raw signals from the CO_{2} absorption and reference bands, X_{wc} is the cross sensitivity coefficient for water on CO_{2} (CO_{2} XS on the calibration sheet), A_{w} and A_{wo} are the raw signals for the water vapor absorption and reference bands, Z_{co} is the zeroing parameter (CO2 Zero on the calibration sheet), Z_{c} is the temperature drift coefficient (CO_{2} Z on the calibration sheet), and V_{c} is the cooler voltage.
Mole density of CO_{2}, ρ_{c}, is given by
The coefficients for the 5th order polynomial f_{c}() are given on the calibration sheet. The polynomial has the form Ax + Bx^{2} + Cx^{3} + Dx^{4} + Ex^{5}, where x = a_{c}S_{c}/P_{ec}. S_{c} is the span parameter for CO_{2}, P_{ec} is equivalent pressure, and a_{c} is the span-drift corrected absorptance for CO_{2}.
The value the LI-7550 needs to output for CO_{2} absorptance is a_{c} (equation 11‑17), and for H_{2}O absorptance is a_{w} (equation 11‑13). The span drift correction, implemented in this manner, should leave the span setting algorithms unchanged.
LI-7500DS implementation
Atmospheric pressure, P_{g}, (kPa) and temperature, T_{g}, (°C) are measured by sensors in the Analyzer Interface Unit. W_{f} is the mole fraction of water vapor and .
Label | Description | Equation |
---|---|---|
H_{2}O mmol/m^{3} | H_{2}O number density | |
H_{2}O g/m^{3} | H_{2}O mass density |
11‑21 |
H_{2}O mmol/mol | H_{2}O mole fraction |
11‑22 |
Dew Point (°C) | Dew point temperature |
11‑23 11‑24 |
CO_{2} mmol/m^{3} | CO_{2} number density | |
CO_{2} mg/m^{3} | CO_{2} mass density |
11‑26 |
CO_{2} µmol/mol | CO_{2} mole fraction |
11‑27 |
A note about pressure and temperature
Since the instrument is calibrated for number density, accurate temperature is not required for the calculation, and accurate pressure measurement is not required, either (equations 11‑20 and 11‑25). For example, if you introduce a 1% error in the pressure sensor on a perfectly calibrated instrument, the resulting CO_{2} mole density error would be about 0.25%, and the H_{2}O mole density error about 0.5% in typical ambient conditions.
When calibrating (specifically when setting spans), temperature and pressure are more important. Calibrating with a 1% pressure error will cause the resulting CO_{2} mole density to have a 1% error, but no error in the resulting H_{2}O mole density (because the water span target is computed from dew point, not mole fraction). A 1% error in temperature (3 °C) will cause a 1% error in both CO_{2} and H_{2}O mole density.