# Equation Summary

The LI-830 and LI-850 compute CO_{2} concentrations using an equation of the form

A‑1

where `c` is concentration, `f`() is the calibration function, `a`" is the absorptance, `g` (`a`,`P`) is the pressure correction, `S`(`a`) is the span, and `T` is the temperature (°C) of the gas in the cell, typically 51.5 °C. Absorptance is computed from

A‑2

`a`' is a span corrected absorptance, and `g`(`a`,`P`) is the pressure correction.

A‑3

`S`(`a`) is the span function, and raw absorptance `a` is computed from

A‑4

where `V` and `V _{o}` are the raw detector sample and reference readings, and

`Z`is the zeroing parameter.

Span is a linear function of absorptance.

A‑5

## H_{2}O Equations (LI-850 only)

Absorptance `a _{w}` for water vapor is computed from

A‑6

where `V _{w}` and

`V`are the sample and reference raw detector readings, and

_{wo}`Z`is the zero parameter. The pressure correction for water vapor is an empirical function

_{w}`g`() of absorptance and pressure

_{w}`P`:

A‑7

The value of `P _{o}` is 99 kPa. When the pressure correction is not enabled,

`g`() is simply 1.0. Water vapor concentration

_{w}`W`(mmol mol

^{-1}) is computed from

A‑8

where `f _{w}`(

`x`) is a third order polynomial whose coefficients are given on the calibration sheet.

A‑9

## CO_{2} Equations

The measurement of CO_{2} is a bit more complicated than for H_{2}O because of the influence of water vapor. There is a slight direct cross sensitivity in the CO_{2} signal to H_{2}O. This is measured at the factory and accounted for in the computation of absorptance (equation A‑10). There is also a band broadening effect that is accounted for in the computation of concentration (equation A‑14).

CO_{2} absorptance `a``c` is computed from

where `V _{c}` and

`V`are the raw detector signals for sample and reference,

_{co}`Z`is the CO

_{c}_{2}zero parameter, and

`X`is a cross sensitivity parameter for the effect of water vapor on CO

_{wc}_{2}. Its value is reported on the calibration sheet as

**XS=**.

The empirical pressure correction function `g _{c}`() depends on CO

_{2}absorptance and pressure:

When `P` = `P _{o}`,

`g`() = 1.

_{c}When `P` < `P _{o}`

where `a` = 1.10158, `b` = -6.1217E-3, `c` = -0.266278, `d` = 3.69895, and `z` is the asymptotic value of absorptance, obtained from the calibration coefficients (equation A‑15).

When *P* > *P*_{o}

A‑13

where `X`, `A`, and `B` are computed as in equation A‑11. CO_{2} concentration `C` (µmol mol^{-1}) is computed from

where `f _{c}`(

`x`) is a function whose inverse is a double rectangular hyperbola, and whose coefficients (

`a1`…

`a4`) are given on the calibration sheet.

Solving equation A‑15 for `C` yields the calibration function

A‑16

Where

A‑17

`ψ`(`W`) accounts for band broadening by water vapor.

A‑18

The band broadening coefficient `h`(`a`_{c}) has been determined to be 1.45 for the instrument for CO_{2} concentrations near ambient. At higher concentrations, the value decreases. We capture this behavior with an empirical relationship (equation A‑19).

Where `z` is from equation A‑12, and `b _{w}` is the low concentration band broadening coefficient: 1.45. This is the value shown on the calibration sheet as

`BB`= 1.45. The typical relationship between

`h`(

`a`) and CO

_{c}_{2}concentration is shown in Figure A‑1. (‘Typical’ because the exact relationship depends on the relationship between absorptance and CO

_{2}, which is the calibration curve.)

**Note:** We formulated equation A‑19 with 0.64`b _{w}` – 0.64 instead of the simple equivalent (0.29) because this allows band broadening corrections to be turned off by setting

`b`to 1. When

_{w}`b`=1,

_{w}`h`(

`a`) = 1 everywhere. Also, to avoid computational problems (underflows, overflows, and division by zero) we constrain the argument

_{c}`a`when computing

_{c}`h`(

`a`) to be 0.1 <

_{c}`a`≤

_{c}`z`.

`a`- 0.1 is typically equivalent to about 600 ppm.

_{c}## Calibration Equations

The following equations describe the implementation of zero and span calibrations.

### Zeroing H_{2}O (LI-850 only)

When the command for zeroing water is received, the LI-850 computes the water zero from equation A‑20, where and are averaged for 5 seconds.

### Zeroing CO_{2}

When the command for zeroing CO_{2} is received, the instrument computes the CO_{2} zero term from equation A‑21, where , , , and are averaged for 5 seconds.

### Spanning H_{2}O (LI-850 only)

When the command for setting the span for H_{2}O is received, along with the target concentration `W _{T}`, from the target concentration, the target absoprtance

`a`is computed from

_{T}A‑22

LI-850 computes `S _{w0}` from equation A‑23 , where is averaged over five seconds.

where

The instrument retains the following values, which are used for subsequent secondary spans:

A‑25

### Secondary Span H_{2}O (LI-850 only)

When the secondary span command for H_{2}O is received, the instrument computes new values for both `S _{w0}` and

`S`. First, it measures a new and computes a new from equation A‑24. Then, it uses these plus the retained values ( and from the previous normal span) to compute

_{w1}A‑26

Given the new span slope `S _{w1}`, it updates the span offset

`S`by equation A‑23.

_{w0}### Spanning CO_{2}

When the command for setting the CO_{2} span is received, along with the target concentration `C _{T}`, the instrument computes

`S`from equation A‑28, where and are averaged for 5 seconds.

_{c0}A‑27

where

Note that

We need `a``cT` to compute , but `a``cT` depends on . We resolve this by using an approximation (equation A‑31) instead when computing equation A‑30

The instrument retains the following values, which are used for subsequent secondary spans, if necessary:

A‑32

A‑33

### Secondary Span CO_{2}

When the secondary span command for CO_{2} is received, the instrument computes new values for both `S _{c0}` and

`S`. First, it measures a new and computes a new

_{c1}`β`from equation A‑29. Then it uses these, plus the retained values (

_{c}`a`

_{c1}and

`β`from the previous normal span) to compute

_{c1}A‑34

Given the new span slope `S _{c1}`, it updates the span offset

`S`by equation A‑28.

_{c0}