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

11‑1

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_iλ. Substituting this into equation 11‑1, and solving for the number density ρ_i of gas i yield

11‑2

We rewrite equation 11‑2 as

11‑3

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

11‑6

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.

11‑7

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):

11‑8

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 CO₂ and H₂O, 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

11‑9

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

11‑10

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

H₂O

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

H₂O absorptance, a_w, is

11‑13

where b₁, b₂, and b₃ are constants (CO₂ 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 CO₂ on water vapor (H₂O XS on the calibration sheet), A_c and A_co are the raw signals for the CO₂ absorption and reference bands, Z_wo is the zeroing coefficient (H2O Zero on the calibration sheet), Z_w is the zero drift coefficient (H₂O Z on the calibration sheet), and V_c is the cooler voltage.

Mole density of H₂O, 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 + Bx² + Cx³, where x = a_wS_w/P. S_w is span for H₂O.

CO₂

The absorption of radiation by CO₂ molecules is influenced by several other gases, including O₂ and H₂O. Since the concentration of H₂O 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.

CO₂ absorptance, a_c, is given by

11‑17

where b₁, b₂, and b₃ are constants (H₂O 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 CO₂ absorption and reference bands, X_wc is the cross sensitivity coefficient for water on CO₂ (CO₂ 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 (CO₂ Z on the calibration sheet), and V_c is the cooler voltage.

Mole density of CO₂, ρ_c, is given by

11‑19

The coefficients for the 5th order polynomial f_c() are given on the calibration sheet. The polynomial has the form Ax + Bx² + Cx³ + Dx⁴ + Ex⁵, where x = a_cS_c/P_ec. S_c is the span parameter for CO₂, P_ec is equivalent pressure, and a_c is the span-drift corrected absorptance for CO₂.

The value the LI-7550 needs to output for CO₂ absorptance is a_c (equation 11‑17), and for H₂O 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 .

Table 11‑1. Fundamental equations used in the LI-7500DS calculations.

Label	Description	Equation
H2O mmol/m3	H2O number density	11‑20
H2O g/m3	H2O mass density	11‑21
H2O mmol/mol	H2O mole fraction	11‑22
Dew Point (°C)	Dew point temperature	11‑23 11‑24
CO2 mmol/m3	CO2 number density	11‑25
CO2 mg/m3	CO2 mass density	11‑26
CO2 µmol/mol	CO2 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 CO₂ mole density error would be about 0.25%, and the H₂O 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 CO₂ mole density to have a 1% error, but no error in the resulting H₂O 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 CO₂ and H₂O mole density.