Accuracies of field CO 2 −H 2 O data from open-path eddy-covariance flux systems: Assessment based on atmospheric physics and biological environment

. Ecosystem CO 2 − H 2 O data measured vastly from open-path eddy-covariance (OPEC) systems by infrared analyzers have numerous applications in biogeosciences. To assess the applicability, data uncertainties from measurements are needed. The uncertainties are sourced from infrared analyzers in zero drift, gain drift, cross-sensitivity, and precision 15 variability. The sourced uncertainties are individually specified for analyzer performance, but no methodology exists to comprehend these individual uncertainties into a cumulative error for the specification of an overall accuracy, which is ultimately needed. Using the methodology for close-path eddy-covariance systems, this accuracy for OPEC systems is determined from all individual uncertainties via an accuracy model further formulated into CO 2 and H 2 O accuracy equations. Based on atmospheric physics and the biological environment, these equations are used to evaluate CO 2 accuracy (±1.21 20 mgCO 2 m − 3 , relatively ±0.19%) and H 2 O accuracy (±0.10 gH 2 O m − 3 , relatively ±0.18% in saturated air at 35 °C and 101.325 kPa). Cross-sensitivity and precision variability are minor, although unavoidable, uncertainties. Zero drifts and gain drifts are major uncertainties but are adjustable via corresponding zero and span procedures during field maintenance. The equations provide rationales to assess and guide the procedures. In an atmospheric CO 2 background, CO 2 zero and span procedures can narrow CO 2 accuracy by 40%, from ±1.21 to ±0.72 mgCO 2 m − 3 . In hot and humid weather, H 2 O gain drift potentially adds 25 more to H 2 O measurement uncertainty, which requires more attention. If H 2 O zero and span procedures can be performed practically from 5 to 35 ºC, the poorest H 2 O accuracy can be improved by 30%, from ±0.10 to ±0.07 gH 2 O m − 3 . Under freezing conditions, an H 2 O span is both impractical and unnecessary, but the zero procedure becomes imperative to minimize H 2 O measurement uncertainty. In cold/dry conditions, the zero procedure for H 2 O, along with CO 2 , is an operational and efficient option to ensure and improve H 2 O accuracy.


Introduction
Open-path eddy-covariance (OPEC) systems are used most to measure boundary-layer CO2, H2O, heat, and momentum fluxes between ecosystems and the atmosphere (Lee and Massman, 2011). For the fluxes, an OPEC system is equipped with a fast-response three-dimensional (3-D) sonic anemometer, to measure 3-D wind and sonic temperature (Ts), and a fastresponse infrared CO2−H2O analyzer (hereafter referred to as an infrared analyzer or analyzer) to measure CO2 and H2O 35 fluctuations (Fig. 1). In this system, the analyzer is adjacent to the sonic measurement volume. Both anemometer and analyzer together provide high-frequency (e.g., 10 to 20 Hz) measurements, which are used to compute the fluxes at a location represented by the measurement volume . The degree of exactness for each flux from computations depends on the field measurement exactness of variables such as CO2, H2O, Ts, and 3-D wind (Foken et al., 2012). Beyond flux computations, the data for individual variables from these field measurements are important in numerous 40 applications. Knowledge of measurement exactness is required for data analysis and applicability assessment (Csavina et al., 2017;Hill et al., 2017). needs rederivation and equations needs reformulation. Following the methdology of Zhou et al. (2021) and using the 80 specifications of EC150 infrared analyzers in OPEC systems as an example (Campbell Scientific Inc., 2021b), we derive the model and formulate equations to assess the accuracies of CO2 and H2O measurements from OPEC systems by infrared analyzers, discuss the uses of accuracies in data applications and analyzer field maintenance, and ultimately provide an reference for the flux measurement community to specify the overall accuracy of field CO2/H2O measurements from OPEC systems by infrared analyzers. 85

Specification implications
An OPEC system for this study includes, but is not limited to, a CSAT3A sonic anemometer for a fast response to 3-D wind and Ts, and an EC150 infrared analyzer for a fast response to CO2 and H2O (Fig. 1). The system operates in a Ta range from -30 to 50 °C and in a P range from 70 to 106 kPa. Within both ranges, the specifications for CO2 and H2O measurements (Campbell Scientific Inc., 2021b) are given in Table 1. 90 0.004 gH2O m -3 a 0.10% is the CO2 gain drift percentage denoted by δCO2_g in text, and ρCO2 is CO2 density. b 0.30% is the H2O gain drift percentage denoted by δH2O_g in text, and ρH2O is H2O density.
In Table 1, the top limit of 1,553 mgCO2 m −3 in the calibration range for CO2 density in dry air is more than double 95 the atmospheric background CO2 density of 760 mgCO2 m −3 , equal to 415 μmolCO2 mol −1 , where mol is for a dry air unit, https://doi.org/10.5194/gi-2022-1 Preprint. Discussion started: 26 January 2022 c Author(s) 2022. CC BY 4.0 License.
reported by Global Monitoring Laboratory (2021) with a Ta of 20 °C under a P of 101.325 kPa (i.e., normal temperature and pressure - Wright et al. (2003)). The top limit of 44 gH2O m −3 in the calibration range for H2O density is equivalent to a 37 °C dew point, higher than the highest 35 °C dew point ever recorded under natural conditions on the Earth (National Weather Service, 2021). 100 The measurement uncertainties of infrared analyzers for CO2 and H2O in Table 1 are specified by individual uncertainty components along with their magnitudes: zero drift, gain drift, cross-sensitivity to CO2/H2O, and precision variability. Zero drift uncertainty is an analyzer non-zero response to zero air/gas (i.e., air/gas free of CO2 and H2O). Gain drift uncertainty is an analyzer trend-deviation response to measured gas species away from its true value in proportion (Campbell Scientific Inc., 2021b). Cross-sensitivity is an analyzer background response to either CO2 if H2O is measured, or 105 H2O if CO2 is measured. Precision variability is an analyzer random response to minor unexpected factors. For CO2 and H2O, respectively, these four components should be composited as a cumulative uncertainty to evaluate the accuracy that is ultimately needed in applications.
Precision variability is a random error, and the other specifications can be considered as trueness. Zero drifts are impacted more by Ta, and so are gain drifts. Additionally, each gain drift is also positively proportional to the true magnitude 110 of CO2/H2O density (i.e., true ρCO2 or true ρH2O) under measurements. Lastly, cross-sensitivity to H2O/CO2 is related to the background amount of H2O/CO2 as indicated by its units, mgCO2 m −3 (gH2O m −3 ) −1 for CO2 measurements, and gH2O m −3 (mgCO2 m −3 ) −1 for H2O measurements.
Accordingly, beyond statistical analysis, the accuracy of CO2/H2O measurements should be evaluated over a Ta range of −30 to 50 °C, a ρH2O range of up to 44 gH2O m −3 , and a ρCO2 range of up to 1,553 mgCO2 m −3 . 115

Accuracy model
The measurement accuracy of infrared analyzers is the maximum range of cumulative measurement uncertainty from the four components uncertainties as specified in Table 1: zero drift, gain drift, cross-sensitivity, and precision variability. The four uncertainties interactionally or independently add uncertainties to a measurement value. Given the true α density (ραT, where subscript α can be either CO2 or H2O) and measured α density (ρα), the difference between the true and measured α 120 densities (Δρα) is given by The analyzer overestimates the true value if Δρα > 0, exactly estimates the true value if Δρα = 0, and underestimates the true value if Δρα < 0. The measurement accuracy is the maximum range of overestimation or underestimation, being a range of Δρα (i.e., an accuracy range). According to the analyses of Zhou et al. (2021) for CPEC infrared analyzers, this range is 125 interactionally contributed by the zero drift uncertainty ( ) ∆ρ α z , gain drift uncertainty ( ) ∆ρ α g , and cross-sensitivity uncertainty ( ) ∆ρ α s while being independently added by the precision uncertainty ( ) ∆ρ α contribution from a pair of uncertainties is three orders smaller in magnitude than each in the pair. The contribution of interactions to the accuracy range can be reasonably neglected. Therefore, the accuracy range can be modeled as a simple sum of the four components uncertainties. From Eq. (A7) in Appendix A, the measurement accuracy of α density from 130 OPEC systems by infrared analyzers is defined in an accuracy model as Assessment on the accuracy of field CO2 or H2O measurements is to formulate and evaluate the four terms on the right side of this accuracy model.

Accuracy of CO2 density measurements 135
Accuracy Model (2) defines the accuracy of field CO2 measurements from OPEC systems by infrared analyzers (ΔρCO2) as where ∆ρ CO z 2 is CO2 zero drift uncertainty, ∆ρ CO g 2 is CO2 gain drift uncertainty, ∆ρ CO s 2 is cross-sensitivity-to-H2O uncertainty, and ∆ρ CO p 2 is CO2 precision uncertainty.
CO2 precision is the standard deviation of ρCO2 random errors among repeated measurements under the same 140 conditions (Joint Committee for Guides in Metrology, 2008). Therefore, using this deviation, the precision uncertainty for an individual CO2 measurement at a 95% confidence interval (P-value of 0.05) can be statistically formulated as (Hoel, 1984) ∆ρ σ CO p CO The other uncertainties, due to CO2 zero drift, CO2 gain drift, and cross-sensitivity-to-H2O, are caused by the inability of the working equation inside an infrared analyzer to be adapted to the changes in its internal and ambient 145 environmental conditions, such as internal housing CO2 and/or H2O levels and ambient air temperature. According to LI−COR Biosciences (2021b), a general model of the working equation for ρCO2 is given by where subscripts c and w indicate CO2 and H2O, respectively; aci (i = 1, 2, 3, 4, or 5) is a coefficient of the five-order polynomial in the terms inside curly brackets; Acs and Aws are the power values of analyzer source lights in the wavelengths 150 for CO2 and H2O measurements, respectively; Ac and Aw are their respective remaining power values after the source lights pass through the measured air; Sw is cross-sensitivity of the detector to H2O, while detecting CO2, in the wavelength for CO2 measurements (hereafter referred to as sensitivity-to-H2O); Zc is the CO2 zero adjustment (i.e., CO2 zero coefficient); and Gc is the CO2 gain adjustment (i.e., commonly known as the CO2 span coefficient). For an individual analyzer, the parameters aci, Zc, Gc, and Sw in Model (5) are statistically estimated in the production calibration against a series of standard CO2 gases at different concentration levels over the ranges of ρH2O and P (hereafter referred to as calibration). The estimated parameters are specific for the analyzer; therefore, Model (5) with these estimated parameters becomes an analyzer-specific CO2 working equation. The working equation is used inside the infrared analyzer to compute ρCO2 from field measurements of Ac, Acs, Aw, Aws, and P.
The analyzer-specific working equation is deemed accurate immediately after calibration (LI−COR Biosciences, 160 2021b). However, as used inside an optical instrument under changing environments vastly different from its manufacturing conditions, the working equation may not be fully adaptable to the changes, which might be reflected through CO2 zero and/or gain drifts of the infrared analyzers in measurements. In the working equation for ρCO2 from Model (5) An infrared analyzer was calibrated for zero air/gas to report zero ρCO2 plus an unaviodable random error. However, during use of the analyzer in measurement environments that are different from calibration conditions, the analyzer often gradually reports this zero ρCO2 value that is different from zero and possibly beyond 2 p CO ρ ±∆ , which is known as CO2 zero drift. This 170 drift is primarily affected by the temperature surrounding the analyzer away from the calibration temperature and/or by traceable CO2 and H2O accumulations during use inside the analyzer light housing due to an inevitable, although extremely little, leaking exchange of housing air with ambient air (hereafter referred to as housing CO2−H2O accumulation). The light housing is technically sealed to keep housing air close to zero air by implementing scrubber chemicals into the housing to absorb any CO2 and H2O that may sneak into the housing through an exchange with any ambient air (LI−COR Biosciences, 175 2021b).
Due to the CO2 zero drift, the working equation needs to be adjusted through its parameter re-estimation to adapt the ambient air temperature and housing CO2−H2O accumulation near which the system is running. This adjustment technique is the zero procedure, which brings the ρCO2 and ρH2O of zero air/gas from the working equation back to zero as closely as possible. In this section, ignoring ρH2O, the discussion focus will be on CO2, although applicability to H2O also exists. In the 180 field, the zero procedure must be simple. The simplest way is to use one air/gas benchmark to re-estimate one parameter in the working equation. This parameter must be adjustable to output zero ρCO2 from the zero air/gas benchmark. It is Zc indeed (see Model (5)), being adjustable to result in a zero ρCO2 value for zero air/gas, if re-estimated for the working equation from where Ac0 and Aw0 are the counterparts of Ac and Aw for zero air/gas, respectively. Inside an analyzer, the zero procedure for CO2 is to re-estimate Zc to balance Eq. (6).
If Zc could continually balance Eq. (6) after the zero procedure, the CO2 zero drift would not be significant; however, this is not the case. Similar to its performance after calibration, an analyzer may still drift after the zero procedures due to changing ambient air temperature and/or CO2−H2O accumulation. Nevertheless, the value of Zc, which should be used 190 with the ambient air temperature surrounding the infrared analyzer and particularly with housing CO2−H2O accumulation, is unpredictable. Assuming that the scrubber chemicals inside the analyzer light housing is replaced as per the recommended maintenance schedule, the housing CO2−H2O accumulation should not be a concern while the air temperature surrounding the infrared analyzer is not controlled. Therefore, the CO2 zero drift of analyzers is specified to be influenced more by Ta and to be ±0.55 mgCO2 m −3 as its maximum range within the operational ranges in Ta and P of OPEC systems (Table 1). This 195 specification is the maximum range of CO2 measurement uncertainty due to the CO2 zero drift.
Given that an analyzer performs best, even without zero drift, at the ambient air temperature for the calibration/zeroing procedure (Tc), and that it possibly drifts while Ta gradually changes away from Tc, then the further away Ta is from Tc, the more it possibly drifts. Over the operational range in P of OPEC systems, this drift is more proportional to the difference between Ta and Tc but is still within the specifications (Campbell Scientific Inc., 2021b

T T T T T T T T T T
where, over the operational range in Ta of OPEC systems, Trh is the highest-end value (50 °C) and Trl is the lowest-end value (-30 °C,

(CO2 gain drift uncertainty)
An infrared analyzer was also calibrated against a series of standard CO2 gases. The calibration sets the working equation from Model (5) to closely follow the gain trend of change in ρCO2. As was determined with the zero drift, the analyzer, with changes in internal CO2−H2O accumulation and ambient measurement conditions during use, could report CO2 gradually away from the real gain trend of the change in ρCO2, which is specifically termed CO2 gain drift. This drift is affected by 210 almost the same factors as the CO2 zero drift (LI−COR Biosciences, 2021b). https://doi.org/10.5194/gi-2022-1 Preprint. Discussion started: 26 January 2022 c Author(s) 2022. CC BY 4.0 License.
Due to the gain drift, the infrared analyzer needs to be further adjusted, after the zero procedure, to tune its working equation back to the real gain trend in ρCO2 of measured air as close as possible. This is done with the CO2 span procedure.
Like the zero procedure, this procedure is simplified by the use of one CO2 span gas, as a benchmark, with a known CO2 amount (ρ CO 2 ) around the typical CO2 density values in the measurement environment. Also, because one CO2 value from 215 CO2 span gas is used, only one parameter in the working equation is available for adjustment. Weighing the gain of the working equation more than any other parameter, this parameter is the CO2 span coefficient (Gc) (see Model (5)). The CO2 span is used to re-estimate Gc to satisfy the following equation (for details, see LI−COR Biosciences, 2021b) Similar to the zero drift, the CO2 gain drift continues after the CO2 span procedure. Based on a similar consideration 220 for the specifications of CO2 zero drift, the CO2 gain drift is specified by the maximum CO2 gain drift percentage (δCO2_g = 0.1%) associated with ρCO2 as ±0.10%×(true ρCO2) ( Table 1). This specification is the maximum range of CO2 measurement uncertainty due to the CO2 gain drift within the operational ranges in Ta and P of OPEC systems.
Given that an analyzer performs best, almost without gain drift, at the ambient air temperature for calibration/span procedure (also denoted by Tc, because zero and span procedures should be performed under similar ambient air temperature 225 conditions) but also drifts while Ta gradually changes away from Tc, then the further away Ta is from Tc, the greater potential the drift has. Accordingly, the same approach to the formulation of CO2 zero drift uncertainty can be applied to the formulation of approximate equation for CO2 gain drift uncertainty at Ta as where ρCO2T is true CO2 density unknown in measurement. Given that the measured value of CO2 density is represented by 230 ρCO2, by referencing Eq. (1), ρCO2T can be expressed as The term inside the parentheses in this equation is the measurement error for ρCO2T that is reasonably smaller in magnitude, by at least two orders, than ρCO2T, whose magnitude in atmospheric background under the normal temperature and pressure as used by Wright et al. (2003) is 760 mgCO2 m -3 (Global Monitoring Laboratory, 2021). Therefore, ρCO2 in Eq. (10) is the best 235 alternative, with the most likelihood, to ρCO2T for the application of Eq. (9). As such, ρCO2T in Eq. (9) can be reasonably approximated by ρCO2 for equation applications. Using this approximation, Eq. (9) becomes ∆ρ δ ρ

T T T T T T T T T T
With measured ρCO2, this equation is applicable in estimating the CO2 gain drift uncertainty. The gain drift uncertainty ) from this equation has the maximum range of ±δCO2_g ρCO2, as if Ta and Tc were separately at the two ends of 240 operational range in Ta of OPEC systems. With the most likelihood, this maximum range is the closest to ±δCO2_g×(true ρCO2) as specified in Table 1.

Sw and ∆ρ CO
s 2 (sensitivity-to-H2O uncertainty) The infrared wavelength of 4.3μm for CO2 measurements is minorly absorbed by H2O (LI−COR Biosciences, 2021b; Campbell Scientific Inc., 2021b). This minor absorption slightly interferes with the absorption by CO2 in the wavelength 245 (McDermitt et al., 1993). The power of the same measurement light through several gas samples with the same CO2 density, but different backgrounds of H2O densities, is detected with different values of Ac for the working equation from Model (5).
Without parameter Sw and its joined term in the working equation, different Ac values must result in significantly different ρCO2 values, although they are actually the same. To report the same ρCO2 for air flows with the same CO2 density under different H2O backgrounds, the different values of Ac to report similar ρCO2 are accounted for by Sw associated with Aw and 250 Aws in the working equation from Model (5). Similar to Zc and Gc in the equation, Sw is not perfectly accurate and can have uncertainty in the determination of ρCO2. This uncertainty for EC150 infrared analyzers is specified by sensitivity-to-H2O (sH2O) as ±2.69×10 -7 mgCO2 m -3 (gH2O m -3 ) -1 ( Table 1). As indicated by its unit, this uncertainty is linearly related to ρH2O.
Assuming the analyzer for CO2 works best, without this uncertainty, in dry air, ∆ρ CO (14) 260 This is the CO2 accuracy equation for the OPEC systems with infrared analyzers. It expresses the accuracy of a field CO2 measurement from the OPEC systems in terms of its specifications σCO2, sH2O,dcz,δCO2_g, and the OPEC system operational range in Ta as indicated by Trh and Trl; measured variables ρCO2 and Ta; and a known variable Tc. Given the specifications and the known variable, this equation can be used to evaluate the CO2 accuracy as a range in relation to Ta and ρCO2.

Evaluation of ΔρCO2 265
Given the analyzer specifications, the accuracy of field CO2 measurements from an infrared analyzer after calibration, zero, and/or span at Tc can be evaluated using the CO2 accuracy equation (14) over a domain of Ta and ρCO2. To visualize the relationship of accuracy with Ta and ρCO2, the accuracy is presented better as the ordinate along the abscissa of Ta for ρCO2 at different levels and must be evaluated within possible maximum ranges of Ta and ρCO2 in ecosystems. In evaluation, the Ta is limited to the -30 to 50 °C range within which OPEC systems operate, Tc can be assumed to be 20 ºC (i.e., standard air 270 temperature as used by Wright et al. (2003)), and ρCO2 can be ranged acoording to its variation in ecosystems.

ρCO2 range
CO2 density measured by the infrared analyzers ranges up to 1,553 mgCO2 m −3 . In the atmosphere, its CO2 background mixing ratio currently is 415 µmolCO2 mol −1 (Global Monitoring Laboratory, 2021). Under the normal temperature and pressure conditions (Wright et al., 2003), this background mixing ratio is equivalent to 760 mgCO2 m -3 in dry air. CO2 275 density in ecosystems commonly ranges from 650 to 1,500 mgCO2 m −3 (LI−COR Biosciences, 2021b), depending on biological processes (Wang et al., 2016), aerodynamic regimes (Yang et al., 2007), and thermodynamic states (Ohkubo et al., 2008). In this study, this range is extended from 600 to 1,600 mgCO2 m -3 as a common range within which ΔρCO2 is evaluated. Because of the dependence of ΔρCO2 on ρCO2 (Eq. 14), to show the accuracy at different CO2 levels, the range is further divided into five grades of 600, 760 (atmospheric background), 1000, 1300, and 1600 mgCO2 m −3 for evaluation 280 presentations as in Fig. 2.
According to the brief review by Zhou et al. (2021) on the plant physiological threshold in air temperature for growth and development and the soil temperature dynamic related to CO2 from microorganism respiration and/or wildlife activities in terrestrial ecosystems, ρCO2 at any grade of 1,000, 1300, or 1600 mgCO2 m −3 should start, at 5 ºC, to converge asymptotically to the atmospheric CO2 background (760 mgCO2 m −3 at -30 ºC, Fig. 2). Without the asymptotical function 285 for the convergence curve, conservatively assuming the convergence has a simple linear trend with Ta from 5 to -30 ºC, ΔρCO2 is evaluated up to the magnitude of ρCO2 along the trend (Fig. 2).

ΔρCO2 range
At Ta = Tc, the CO2 accuracy is best at its narrowest range as the sum of precision and and sensitivity-to-H2O uncertainties (±0.39 mgCO2 m −3 ). However, away from Tc, its range near-linearly becomes wider. The ΔρCO2 range can be summarized as 290  Table 2). The maximum CO2 relative accuracy at the different levels of ρCO2 is in a range of ±0.07% at 1,600 mgCO2 m −3 to 0.19% at 600 mgCO2 m −3 (from data for Fig. 2b).  analyzers over their operational range in Ta at atmospheric pressure of 101.325 kPa. The vertical dashed line represents ambient temperature Tc at which an analyzer was calibrated, zeroed, and/or spanned. Above 5 °C, accuracy is evaluated up to the possible maximum CO2 density in ecosystems (black curve). Assume this maximum starts linearly decreasing at 5 °C to the atmospheric CO2 background (760 mgCO2 m -3 ) at -30 °C. Accordingly, below 5 °C, the accuracy for CO2 density at a 300 level above the background (green, blue, or black curve) is evaluated up to this decreasing trend. Relative accuracy of CO2 measurements is the ratio of CO2 accuracy to CO2 density.  . 310 e H2O density in air of 60% relative humidity above 48 °C is out of the measurement range of EC150 infrared CO2−H2O analyzers (0 − 44 gH2O m -3 ).

Accuracy of H2O density measurements
Model (2) defines the accuracy of field H2O measurements from OPEC systems by infrared analyzers (ΔρH2O) as The other uncertainty terms in Model (15) where awi (i = 1, 2, or 3) is a coefficient of the three-order polynomial in the terms inside curly brackets; Sc is the cross-325 sensitivity of a detector to CO2, while detecting H2O, in the wavelength for H2O measurements (hereafter referred to as sensitivity-to-CO2); Zw is the H2O zero adjustment (i.e., H2O zero coefficient); and Gw is the H2O gain adjustment (i.e., commonly referred as to H2O span coefficient). The parameters of awi, Zw Gw, and Sc in Model (17) are statistically estimated to establish an H2O working equation in production calibration against a series of air standards with different H2O contents under ranges of ρCO2 and P (i.e., calibration). The H2O working equation (i.e., Model 17 with estimated parameters) is used 330 inside the analyzer to compute ρH2O from field measurements of Aw, Aws, Ac, Acs, and P.
Because of the similarity in model principles and parameter implications between Models (5) and (17)

∆ρ H O
s 2 (sensitivity-to-CO2 uncertainty) The infrared light at wavelength of 2.7 μm for H2O measurement is traceably absorbed by CO2 (see Fig. 4.7 in Wallace and Hobbs, 2006). This absorption interferes slightly with the absorption by H2O in the wavelength (McDermitt et al., 1993). As  ρH2O and Ta; and a known variable Tc. Using this equation and the system specification values in Table 1, the accuracy of field H2O measurements can be evaluated as a range. 360

Evaluation of ΔρH2O
H2O accuracy (ΔρH2O) can be evaluated using the H2O accuracy equation over a domain of Ta and ρH2O. Similar to the CO2 accuracy equation in Fig. 2, ΔρH2O is presented as the ordinate along the abscissa of Ta at different ρH2O levels within the ranges of Ta and ρH2O in ecosystems (Fig. 3). As with the evaluation of ΔρCO2, Ta is limited from -30 to 50 °C and Tc can be assumed to be 20 ºC. The range of ρH2O at Ta needs to be be determined using atmospheric physics (Buck, 1981). 365

ρH2O range
The analyzers measure H2O density from 0 to 44 gH2O m -3 . However, due to the positive exponential dependence of air water vapor saturation on Ta (Wallace and Hobbs, 2006), ρH2O has a range that is wider at higher Ta and narrower at lower Ta.
Below 37 ºC at 101.325 kPa, ρH2O is lower than 44 gH2O m -3 , and its range becomes narrower and narrower, reaching 0.34 gH2O m -3 at -30 ºC. To determine the H2O accuracy over the same relative range of air moisture, even at different Ta, the 370 water vapor saturation density is used to scale air moisture to 20, 40, 60, 80 and 100% (i.e., relative humidity, or RH). For each scaled RH value, ρH2O can be calculated at different Ta and P (Appendix B) for use in the H2O accuracy equation. In this way, over the range of Ta, H2O accuracy can be shown as curves with equal RH (Fig. 3).

ΔρH2O range
In the same way as with CO2 accuracy, the H2O accuracy at Ta = Tc is best at its narrowest as the sum of precision and 375 sensitivity-to-CO2 uncertainties (<0.040 gH2O m −3 in magnitude). However, away from Tc, its non-linear range becomes wider, very gradually below this Tc value but more abruptly above, because, as Ta increases, ρH2O at the same RH increases exponentially (Eqs. B1 and B2 in Appendix B) while ΔρH2O increases linearly with ρH2O in the H2O accuracy equation (22).
This non-linear range can be summarized as the widest at 48 °C to be ±0.099 gH2O m -3 for air with 60% RH (Fig. 3a and H2O columns in Table 2). The number can be rounded up to ±0.10 gH2O m -3 for the overall accuracy of field H2O 380 measurements from OPEC systems by the EC150 infrared analyzers. Fig. 3b shows an interesting trend of H2O relative accuracy with Ta. Given the RH range shown in Fig. 3b, the relative accuracy diverges with a Ta decrease and converges with a Ta increase. The H2O relative accuracy varies from 0.17% for saturated air at 37 ºC to 96% for 20% RH air at -30 ºC (data for Fig. 3b) and, at this low Ta, can be much greater if RH goes further lower. The H2O relative accuracy in magnitude is < 1% while ρH2O > 5.00 gH2O m −3 , < 5% while ρH2O > 1.20 385 gH2O m −3 , and >10% while ρH2O < 0.60 gH2O m −3 .

Discussion
The primary objective of this study is to develop an assessment methodology to evaluate the overall accuracies of field CO2 and H2O measurements from OPEC systems by the infrared analyzers from their individual measurement uncertainties as 395 specified using four uncertainty descriptors: zero drift, gain drift, sensitivity-to-CO2/H2O, and precision variability (Table   1). For the evaluation, these uncertainty descriptors are comprehensively composited into the accuracy model (2) formulated as a CO2 accuracy equation (14) and an H2O accuracy equation (22) (Sects. 3 to 5 and Appendix A). The assessment methodology, along with the model and the equations, is our development for the objective (Sects. 4.5 and 5.4). The evaluated accuracy can be used to assess CO2 and H2O data applications, and the formulated accuracy equations further 400 provide rationales to assess and guide field maintenance on the infrared analyzers.

Methodology development
The methodology is developed from the derivation of accuracy model for the formulation of CO2 and H2O accuracy equations applicable in ecosystems to the evaluation of field CO2 and H2O measurement accuracies.

Accuracy model 405
Accuracy model (2) composites the four measurement uncertainties (zero drift, gain drift, sensitivity-to-CO2/H2O, and precision variability) specified for analyzer performance as an accuracy range. This range is modeled as a simple addition of the four uncertainties. The simple addition is derived from our analysis assertion that the four measurement uncertainties interactionally or independently contribute to the accuracy range, but the contribution from the interaction inside any pair of uncertainties is negligible because the interaction is three orders smaller in magnitude than an individual uncertainty in the 410 pair (Appendix A). This derived model is simple and applicable, opening an approach to the formulation of accuracy equations that are computable to evaluate the overall accuracies of field CO2 and H2O measurements from OPEC systems by infrared analyzers.

Formulation of uncertainty terms in Model (2) for accuracy equations
In Sects. 4 and 5, each of the four uncertainty terms in accuracy model (2) is formulated as a computable sub-equation for 415 CO2 and H2O,respectively (Eqs. 4,7,11,13,16,18,19,or 21). The accuracy model, whose terms are replaced with the formulated sub-equations for CO2, becomes a CO2 accuracy equation and, for H2O, becomes an H2O accuracy equation. In the formulation, approximation is used for zero drift, gain drift, and sensitivity-to-CO2/H2O, while statistics are applied for precision variability.
For the zero/gain drift, although it is well known that the drift is influenced more by Ta if housing CO2−H2O 420 accumulation is assumed to be minimized as insignificant under normal field maintenance (LI−COR Biosciences, 2021b; Campbell Scientific Inc., 2021b), the exact relationship of drift to Ta does not exist. Alternatively, the zero/gain drift https://doi.org/10.5194/gi-2022-1 Preprint. Discussion started: 26 January 2022 c Author(s) 2022. CC BY 4.0 License. uncertainty is formulated by an approximation of drifts away from Tc linearly in proportion to the difference between Ta and Tc but within its maximum range over the operational range in Ta of OPEC systems (Eqs. 7,11,18,and 19). A drift uncertainty equation formulated through such an approximation is not an exact relationship of drift to Ta, but it does 425 represent the drift trend, as influenced by Ta, to be understood by users. The accuracy from this equation at a given Ta is not exact either, but the maximum range over the full range, which is the most likelihood estimation, is most needed by users.
The sensitivity-to-CO2/H2O uncertainty can be formally formulated as Eq. (20) (13). This approximation widens the accuracy range slightly, in a magnitude smaller than each of major uncertainties from the drifts at least in one order; however, it eliminates the need for 435 ρH2O in the CO2 accuracy equation and for ρCO2 in the H2O accuracy equation, which makes the equations easily applicable.
Precision uncertainty is statistically formulated as Eq. (4) for CO2 and Eq. (16) for H2O. This formulation is a common practice based on statistical methods (Hoel, 1984).

Use of relative accuracy for infrared analyzer specifications
Relative accuracy is often used concurrently with accuracy to specify sensor measurement performance. The accuracy is the 440 numerator of relative accuracy whose denominator is the true value of a measured variable. When evaluated for the applications of OPEC systems in ecosystems, CO2 accuracy magnitude is small in a range within one order (0.39 ~ 1.21 mgCO2 m -3 , data for Fig. 2a), and so is H2O accuracy (0.04 ~ 0.10 gH2O m -3 , data for Fig. 3a). In ecosystems, CO2 is naturally high, as compared to its accuracy magnitude, and does not change much in terms of a magnitude order (e.g., no more than one order from 600 to 1,600 gH2O m -3 , assumed in this study). However, unlike CO2, H2O naturally changes in its 445 amount dramatically across at least three orders in magnitude (e.g., at 101.325 kPa, from 0.03 gH2O m -3 when RH is 10% at -30 ºC to 40 gH2O m -3 when dew point temperature is 35 ºC at the highest as reported by National Weather Service (2021); under drier conditions, the H2O amount could be even lower). Because, in ecosystems, CO2 changes differently than H2O in amount across magnitude orders, the relative accuracy behaviors in CO2 differ from H2O (Figs. 2b and 3b).

CO2 relative accuracy 450
Because of the small CO2 accuracy magnitude relative to the natural CO2 amount in ecosystems, the CO2 relative accuracy magnitude varies within a narrow range of 0.07 to 0.19% (Sect. 4.5.2). If the relative accuracy is used, either a range of 0.07 − 0.19% or an inequality of ≤ 0.19% can be specified as the CO2 relative accuracy magnitude for field CO2 measurements. Both range and inequality would be equivalently perceived by users to be a fair performance of OPEC systems. For simplicity, our study with the OPEC systems can be specified for their CO2 relative accuracy to be ±0.19%. 455

H2O relative accuracy
Although the H2O accuracy magnitude is also small, the "relatively" great change in natural air H2O across several magnitude orders in ecosystems results in a much wider range of the H2O relative accuracy magnitude, from 0.23% at maximum air moisture to 96% when RH is 20% at -30 ºC (Fig. 3b and Sect. 5.4.2). H2O relative accuracy can be much greater under dry conditions at low Ta (e.g., 192% for air when RH is 10% at -30 ºC). Accordingly, if the relative accuracy is 460 used, either a range of 0.23 − 192% or an inequality of ≤ 192% can be specified as the H2O relative accuracy magnitude for field H2O measurements. Either 0.23 − 192% or ≤ 192% could be perceived by users intrinsically as poor measurement performance of the infrared analyzers, although either specification is conditionally right for fair H2O measurement.
Apparently, the relative accuracy for H2O measurements in ecosystems is not intrinsically interpretable by users to correctly perceive the performance of OPEC systems. Instead, if H2O relative accuracy is unconditionally specified just in an 465 inequity of ≤ 192%, it could easily mislead users to wrongly assess the performance of OPEC systems as unacceptable for H2O measurements, although this performance of OPEC systems is fair for air when RH is 10% at -30 ºC. Therefore, H2O relative accuracy is not recommended to be used for specification of infrared analyzers for H2O measurement performance.
If this descriptor is used, the H2O relative accuracy under a standard condition should be specified. This condition may be defined as saturated air at 35 ºC (i.e., the highest natural dew point (National Weather Service, 2021)) under normal P of 470 101.325 kPa (Wright et al., 2003). For our study case, under such a standard condition, the H2O relative accuracy can be specified within ±0.18% after manufacturing calibration (data for Fig. 3b).

Application of H2O accuracy in data use
The measured variables ρH2O, Ts and P can be used to compute Ta (Swiatek, 2018 were an exact function from the theoretical principles, it would not have any error itself. However, in our applications, variables ρH2O, Ts, and P are 475 measured from the OPEC systems experiencing seasonal climates. As addressed in this study, the measured values of these variables have measurement uncertainty in ρH2O (ΔρH2O, i.e., accuracy of field H2O measurement); in Ts (ΔTs, i.e., accuracy of field Ts measurement); and in P (ΔP, i.e., accuracy of field P measurement). The uncertainties from measurement propagate to the computed Ta as an uncertainty (ΔTa, i.e., accuracy of T T P a H O s ( , , ) ρ 2 ). This accuracy is a reference by any application of Ta. It should be specified through its relationship of ΔTa to ΔρH2O, ΔTs, and ΔP. 480 As field measurement uncertainties, ΔρH2O, ΔTs, or ΔP are reasonably small increments in numerical analysis (Burden et al., 2016). As such, depending on all the small increments, ΔTa is a total differential of T T P a H O s ( , , ) ρ 2 with respect to ρH2O, Ts, and P, which are measured independently by three sensors, given by https://doi.org/10.5194/gi-2022-1 Preprint. Discussion started: 26 January 2022 c Author(s) 2022. CC BY 4.0 License.
In this equation, ΔρH2O from the application of Eq. (22) is a necessary term to acquire ΔTa, ΔTs can be acquired from the 485 specifications for 3-D sonic anemometers (Zhou et al., 2018), ΔP can be acquired from the specifications for the barometer used in the OPEC systems (Vaisala, 2020), and the three partial derivatives can be derived from the explicit function With ΔρH2O, ΔTs, ΔP, and the three partial derivatives, ΔTa can be ranged as a function of ρH2O, Ts, and P.

Application of accuracy equations in analyzer field maintenance
An infrared analyzer performs better if the field environment is near its manufacturing conditions (e.g., Ta at 20 °C), which is 490 demonstrated in Figs. 2a and 3a for measurement accuracies associated with Tc. As indicated by the accuracies in both figures, the closer to Tc at 20 °C while Ta is, the better analyzers perform. However, the analyzers are mostly used in OPEC systems for long-term field campaigns through four-seasonal climates vastly different from those in the manufacturing processes. Over time, an analyzer gradually drifts in some ways and needs field maintenance although within its specifications. 495 The field maintenance cannot improve the sensitivity-to-CO2/H2O uncertainty and precision variability, but both are minor (their sum < 0.392 mgCO2 m −3 for CO2, Eqs. 4 and 13; < 0.045 gH2O m −3 for H2O, Eqs. 16 and 21) as compared to the zero or gain drift uncertainties. However, the zero and gain drift uncertainties are major in determination of field CO2/H2O measurement accuracy (Figs. 2 to 4 and Eqs. 14 and 22), but adjustable, through the zero and/or span procedures, to be minimized. Therefore, manufacturers of infrared analyzers have provided software and hardware tools for the procedures 500 (Campbell Scientific Inc., 2021b) and scheduled the procedures using those tools (LI−COR Biosciences, 2021b). This study provides rationales how to assess, schedule, and perform the procedures (Figs. 2a, 3a, and 4). Figure 4a shows that the CO2 zero drift uncertainty linearly increases with Ta away from Tc over the full Ta range within which OPEC systems operate; so, too, does CO2 gain drift uncertainty increase for a given CO2 concentration. As suggested 505 by Zhou et al. (2021), both drifts should be adjusted near the Ta value around which the system runs. The zero and gain drifts should be adjusted, through zero and span procedures, at a Ta close to its daily mean around which the system runs. Based on the range of Ta daily cycle, the procedures are set at a moderate, instead of the highest or lowest, moment in Ta. Given the daily cycle range is much narrower than 40 °C, an OPEC system could run at Ta within ±20 of Tc if the procedures are performed at a right moment of Ta. For our study case on atmospheric CO2 background (left CO2 column in Table 2), the 510 procedures can narrow the widest possible range of ±1.21 mgCO2 m −3 for field CO2 measurement at least 40% to ±0.72 mgCO2 m -3 (i.e., accuracy at 0 or 40 ºC when Tc = 20 ºC), which would be a significant improvement to ensure field CO2 measurement accuracy through CO2 zero and span procedures.

CO2 zero and span procedures
https://doi.org/10.5194/gi-2022-1 Preprint. Discussion started: 26 January 2022 c Author(s) 2022. CC BY 4.0 License. Figure 4b shows that the H2O zero drift uncertainty increases as Ta moves away from Tc in the same trend as CO2 zero drift 515 uncertainty. Therefore, an H2O zero procedure can be performed in the same technique as for CO2 zero procedure. H2O gain drift uncertainty has a different trend. It exponentially diverges, as Ta increases away from Tc, to ±5.0 × 10 −2 gH2O m −3 near 50 °C, and gradually converges by two orders smaller, as Ta decreases away from Tc, to ±6.38 × 10 −4 gH2O m −3 at -30 °C (data for Fig. 4b). The exponential divergence results from the linear relationship of H2O gain drift uncertainty (Eq. 19) with ρH2O, which exponentially increases (Eq. B1) with a Ta increase away from Tc for the same RH (Buck, 1981). The 520 convergence results from the linear relationship offset by the exponential decrease in ρH2O with a Ta decrease for the same RH. This trend of H2O gain drift uncertainty with Ta is a rationale to guide the H2O span procedure, which adjusts the H2O gain drift.

H2O zero and span procedures
The H2O span procedure needs standard moist air with known H2O density from a dew point generator. The generator is not operational near or below freezing conditions (LI−COR Biosciences, 2004), which limits the span procedure 525 to be performed only under non-freezing conditions. This condition, from 5 to 35 ºC, may be considered for the generator to be conveniently operational in the field. Accordingly, the H2O zero and span procedures should be discussed separately for a Ta above and below 5 ºC.

Ta above 5 ºC
Looking at the right portion with Ta above 5 ºC in Fig. 4b, H2O gain drift has a more obvious impact on measurement 530 uncertainty in a higher Ta range (e.g., above Tc), within which the H2O span procedure is most needed. In this range, the maximum accuracy range of ±0.10 gH2O m -3 can be narrowed by 30% to ±0.07 (assessed from data for Fig 3a) if H2O zero and span procedures can be sequentially performed as necessary in a Ta range from 5 to 35 ºC.

Ta below 5 ºC
Looking at the left portion with Ta below 5 ºC in Fig 4b, H2O gain drift has a less obvious contribution to the measurement 535 uncertainty in a lower Ta range (e.g., below 5 ºC), within which the H2O span procedure may be unnecessary. An H2O gain drift uncertainty at 5 ºC is 50% of the H2O zero drift uncertainty (dotted curve in Fig. 5). This percentage decreases to 3% at -30 ºC. On average, this percentage over a range of -30 to 5 ºC is 18% (assessed from data for dotted curve in Fig. 5). Thus, for H2O measurements over the lower Ta range, it can be concluded that H2O zero drift is a major uncertainty source, and H2O gain drift is a minor uncertainty source.  A close examination of the other curves in Fig. 5 for the portion in the accuracy range from H2O zero/gain drift 545 makes this conclusion more convincing. Given Tc = 20, in accuracy range, the portion from H2O zero drift uncertainty is much greater (maximum 38% at -30 ºC) than that from H2O gain drift uncertainty (maximum only 7% at 5 ºC). On average over the lower Ta range, the former is 27% and the latter only 4%. Further, given Tc = 5 ºC, in the accuracy range, the portion from H2O gain drift uncertainty is even smaller (maximum only 3% at -5 ºC); in contrast, the portion from zero drift uncertainty is more major (one order higher, 30% at -30 ºC). On average over the lower Ta range, the minor gain drift 550 uncertainty is 1.7%, and the major zero drift uncertainty is 17%. Both percentages underscore that the H2O span procedure is reasonably unnecessary under cold/dry conditions, and, under such conditions, the H2O zero procedure is the only necessary option to efficiently minimize H2O measurement uncertainty in OPEC systems. This finding gives confidence in H2O measurement accuracy to users who are worried about H2O span procedures for infrared analyzers in the cold seasons when a dew point generator is not operational in the field (LI−COR Biosciences, 2004). 555

H2O zero procedure in cold and/or dry environments
In cold environments, although the non-operational H2O span procedure is unnecessary, the H2O zero procedure is asserted to be a prominently important option for minimizing the H2O measurement uncertainty in OPEC systems. This procedure, 565 although operational under freezing conditions, is still inconvenient for users when weather is very cold (e.g., when Ta is below -15 °C). If the field H2O zero procedure is performed as needed above this Ta value, an OPEC system can be assumed to run at Ta with ±20 °C of Tc. Under this assumption, the poorest H2O accuracy of ±0.066 gH2O m -3 below 5 °C in Table 2 can be narrowed, through the H2O zero procedure, by at least 22% to 0.051 gH2O m -3 (assessed from data for Fig. 3a).
Correspondingly, the relative accuracy range can be narrowed by the same percentage. The H2O zero procedure can ensure 570 both accuracy and relative accuracy of H2O measurements in a cold environment. In a dry environment, it plays the same role as in a cold environment, but it would be more convenient for users if warmer.
In a cold and/or dry environment, H2O zero procedures that are undergone on a regular schedule would best minimize the impact of zero drifts on measurements. Under such an environment, the automatic zero procedure for CO2 and H2O together in CPEC systems is an operational and efficient option to ensure and improve field CO2 and H2O measurement 575 accuracies (Campbell Scientific Inc., 2021a; Zhou et al., 2021).

Conclusions
The accuracy of field CO2/H2O measurements from OPEC systems by the infrared analyzers can be defined as a maximum range of composite measurement uncertainty sourced from component uncertainties: zero drift, gain drift, sensitivity-to-CO2/H2O, and precision variability, all of which are included in the system specifications ( Table 1). The specified 580 uncertainties interactionally or independently contribute to the overall uncertainty. Fortunately, the interactions between component uncertainties in each pair is three orders smaller than either component individually (Appendix A). Therefore, these specified uncertainties can be simply added as the accuracy range in a general CO2/H2O accuracy model for OPEC systems (Model 2). Based on statistics, bio-environment, and approximation, the specification descriptors of the infrared analyzers in OPEC systems are incorporated into the model terms to formulate the CO2 accuracy equation (14) Fig. 2) and H2O accuracy as ±0.10 gH2O m -3 (relatively within ±0.18% for saturated air at 35 ºC at the standard P, Fig. 3).
Both accuracy equations are not only applicable for further error/uncertainty analyses in CO2 and H2O data 590 applications (see Sect. 6.1), but they also may be used as a rationale to assess and guide field maintenance on infrared analyzers. Combining Eq. (14) as shown in Fig. 2a with Eqs. (7) and (11) as shown in Fig. 4a guides users to adjust the CO2 zero and gain drifts, through the corresponding zero and span procedures, near the middle of the Ta range within which the analyzer runs. As assessed on atmospheric background, the procedures can narrow the maximum CO2 accuracy range by https://doi.org/10.5194/gi-2022-1 Preprint. Discussion started: 26 January 2022 c Author(s) 2022. CC BY 4.0 License. 40%, from ±1.21 to ±0.72 mgCO2 m -3 , and thereby greatly improve the CO2 measurement accuracies with these regular CO2 595 zero and span procedures.
Equation (22) as shown in Fig. 3a, plus Eqs. (18) and (19) as shown in Fig. 4b, present users with a rationale to adjust the H2O zero drift of analyzers in the same technique as for CO2, but the H2O gain drift under hot and humid environments needs more attention (see the right portion above Tc in Figs. 3a and 4b); under cold and/or dry environments, it needs no further concern (see the left portion below 0 ºC in Fig. 4b). In a Ta range above 5 ºC, the maximum H2O accuracy 600 range of ±0.10 gH2O m −3 can be narrowed by 30% to ±0.07 gH2O m −3 if both H2O zero and span procedures are performed as necessary. In a Ta range below 5 ºC, the H2O zero procedure alone can narrow the maximum H2O accuracy range of ±0.076 gH2O m −3 by 22%, to ±0.051 gH2O m −3 . Under cold environmental conditions, the H2O span procedure is found to be unnecessary (Fig. 5), and the H2O zero procedure is proposed as the only, and prominently efficient, option to minimize H2O measurement uncertainty in OPEC systems. This procedure plays the same role under dry conditions. Under cold and/or dry 605 environments, the zero procedure for CO2 and H2O together would be a practical and efficient option to not only warrant, but also to improve, measurement accuracy. In a cold environment, adjusting the H2O gain drift is impractical because of a dew point generator that fails to generate standard H2O gas near freezing conditions. This lack of necessity relieves user worry with regard to H2O measurement uncertainty from the H2O gain drift under such environments where the H2O span procedure is not operational. 610 Additionally, as a specification descriptor for OPEC systems used in ecosystems, relative accuracy is applicable for CO2 instead of H2O measurements because, in ecosystems, the CO2 relative accuracy varies slightly within a magnitude order, and the H2O relative accuracy varies dramatically across several magnitude orders. A small range in the CO2 relative accuracy can be perceived intuitively by users as normal. In contrast, without specifying the condition of air moisture, a large range in H2O relative accuracy under cold and/or dry conditions (e.g., 100%) can easily mislead users to automatically 615 transfer this relative accuracy onto very poor H2O measurements, although, under such conditions, it is the best that modern technology can do in the field. The authors suggest to conditionally define H2O relative accuracy at 35 °C dew point (i.e., 39.66 gH2O m −3 at 101.352 kPa). Ultimately, this study provides our logic to the flux community in specifying the accuracy of CO2−H2O measurement from OPEC systems by infrared analyzers.

Appendix A: Derivation of accuracy model for infrared CO2−H2O analyzers 620
As defined in the Introduction, the measurement accuracy of infrared CO2−H2O analyzers is a range of the difference between the true α density (ραT, where α can be either H2O or CO2) and measured α density (ρα) by the analyzer. The difference is denoted by Δρα, given by Eq. (1) in Sect. 3. Analyzer performance uncertainties contribute to this range, as specified in the four descriptors: zero drift, gain drift, cross-sensitivity, and precision (LI−COR Biosciences, 2021b; The range of the right side of this equation is wider than the measurement uncertainty from all measurement uncertainty 655 sources and the difference of ρα minus ραT (i.e., Δρα). Using this range, the measurement accuracy is defined in Model (2) in Sect. 3.
Appendix B: Water vapor density from ambient air temperature, relative humidity, and atmospheric pressure Given ambient air temperature (Ta in °C) and atmospheric pressure (P in kPa), air has a limited capacity to hold an amount of water vapor (Wallace and Hobbs, 2006). This limited capacity is described in terms of saturation water vapor density (ρs 660 in gH2O m −3 ) for moist air, given through the Clausius−Clapeyron equation (Sonntag, 1990;Wallace and Hobbs, 2006) ( ) where Rv is the gas constant for water vapor (4.61495 ×10 -4 kPa m 3 K -1 gH2O -1 ), and f(P) is an enhancement factor for moist air, being a function of P: f P P P ( ) . .