CO_dos: CO₂ Correction Tools

Installation

You can(not) install the released version of codos from CRAN with:

install.packages("codos")

And the development version from GitHub with:

# install.packages("devtools")
devtools::install_github("special-uor/codos", "dev")

cat(paste0("
-----

Note: Some of the equations on this document are not displayed properly (due to a server issue), check out the [README-extended.pdf](README-extended.pdf).

-----
"))

Note: Some of the equations on this document are not displayed properly (due to a server issue), check out the README-extended.pdf.

Background:

Vapour-pressure deficit (`vpd`)

vpd is given by mean daily growing season temperature, tmp [°C] and moisture index, mi [-]. Using the CRU TS 4.04 dataset (University of East Anglia Climatic Research Unit et al. 2020) we found the following relation:

$\text{vpd} = 4.6123 \times \exp(0.0609 \times \text{tmp}-0.8726 \times \text{mi})$

The steps performed were:

Generate a monthly climatology for the period between 1961 and 1990 (inclusive). Variables used: cld, pre, tmn, tmx, vap.

# Monthly climatology for `tmn`
codos::monthly_clim("cru_ts4.04.1901.2019.tmn.dat.nc", "tmn", 1961, 1990)

Output file:

"cru_ts4.04.1901.2019.tmn.dat-clim-1961-1990.nc"

Interpolate the monthly data to daily. Variables used: cld, pre, tmn, tmx, vap.

# Monthly to daily interpolation for `tmn`
codos::nc_int("cru_ts4.04.1901.2019.tmn.dat-clim-1961-1990.nc", "tmn")

Output file:

"cru_ts4.04.1901.2019.tmn.dat-clim-1961-1990-int.nc"

Calculate daily temperature, tmp. Variables used: tmn and tmx.

codos::daily_temp(tmin = list(filename = "cru_ts4.04.1901.2019.tmn.dat-clim-1961-1990-int.nc",
                              id = "tmn"),
                  tmax = list(filename = "cru_ts4.04.1901.2019.tmx.dat-clim-1961-1990-int.nc", 
                              id = "tmx"),
                  output_filename = "cru_ts4.04-clim-1961-1990-daily.tmp.nc")

Calculate mean growing season for daily temperature

codos::nc_gs("cru_ts4.04-clim-1961-1990-daily.tmp.nc", "tmp", thr = 0)

Output file:

"cru_ts4.04-clim-1961-1990-daily.tmp-gs.nc"

Calculate potential evapotranspiration (pet)

Install SPLASH (unofficial R package) as follows:

remotes::install_github("villegar/splash", "dev")

Or, download from the official source: https://bitbucket.org/labprentice/splash.

elv <- codos:::nc_var_get("halfdeg.elv.nc", "elv")$data
tmp <- codos:::nc_var_get("cru_ts4.04.1901.2019.daily.tmp.nc", "tmp")$data
cld <- codos:::nc_var_get("cru_ts4.04.1901.2019.cld.dat-clim-1961-1990-int.nc", 
                          "cld")$data

codos::splash_evap(output_filename = "cru_ts4.04-clim-1961-1990-pet.nc", 
                   elv, # Elevation, 720x360 grid 
                   sf = 1 - cld / 100, 
                   tmp, 
                   year = 1961, # Reference year 
                   lat = codos::lat, 
                   lon = codos::lon)

Output file:

"cru_ts4.04-clim-1961-1990-pet.nc"

Calculate moisture index (mi)

$MI_{i,j} = \frac{\text{Total precipitation}}{\text{Total PET}}$

pet <- codos:::nc_var_get("cru_ts4.04-clim-1961-1990-pet.nc", 
                          "pet")$data
pre <- codos:::nc_var_get("cru_ts4.04.1901.2019.pre.dat-new-clim-1961-1990-int.nc", 
                          "pre")$data
codos::nc_mi(output_filename = "cru_ts4.04-clim-1961-1990-mi.nc", 
             pet, # potential evapotranspiration
             pre) # precipitation

Output file:

"cru_ts4.04-clim-1961-1990-mi.nc"

Approximate vpd

tmp <- codos:::nc_var_get("cru_ts4.04-clim-1961-1990-daily.tmp.nc", 
                          "tmp")$data
vap <- codos:::nc_var_get("cru_ts4.04.1901.2019.vap.dat-clim-1961-1990-int.nc", 
                          "vap")$data
output_filename <- file.path(path, "cru_ts4.04-clim-1961-1990-vpd-tmp.nc")
codos::nc_vpd(output_filename, tmp, vap)

Output file:

"cru_ts4.04-clim-1961-1990-vpd-tmp.nc"

Find the coeffients for the following equation

$\text{vpd} = a \times \exp(\text{kTmp} \times \text{tmp}-\text{kMI} \times \text{mi})$

mi <- codos:::nc_var_get("cru_ts4.04-clim-1961-1990-mi.nc", "mi")$data
Tmp <- codos:::nc_var_get("cru_ts4.04-clim-1961-1990-daily.tmp-gs.nc", "tmp")$data
vpd <- codos:::nc_var_get("cru_ts4.04-clim-1961-1990-vpd-tmp-gs.nc", "vpd")$data

# Apply ice mask
mi[codos:::ice_mask] <- NA
Tmp[codos:::ice_mask] <- NA
vpd[codos:::ice_mask] <- NA

# Filter low temperatures, Tmp < 5
mi[Tmp < 5] <- NA
Tmp[Tmp < 5] <- NA

# Create data frame
df <- tibble::tibble(Tmp = c(Tmp), 
                     vpd = c(vpd),
                     MI = c(mi))

# Filter grid cells with missing Tmp, vpd, or MI
df <- df[!is.na(df$Tmp) & !is.na(df$vpd) & !is.na(df$MI), ]

# Linear approximation
lmod <- lm(log(vpd) ~ Tmp + MI, data = df)
# Non-linear model
exp_mod <- nls(vpd ~ a * exp(kTmp * Tmp - kMI * MI),
               df,
               start = list(a = exp(coef(lmod)[1]),
                            kTmp = coef(lmod)[2],
                            kMI = coef(lmod)[3]),
               control = list(maxiter = 200))

Summary statistics:

summary(exp_mod)
coefficients(exp_mod)

Corrected `mi` from reconstructed `mi`

The following equations were used:

$f = e/1.6 = D/[\text{c}_\text{a}(1-\chi)]$

$\chi = \frac{\xi \times \text{vpd}^{1/2} + \text{vpd}}{\text{c}_\text{a}/\left(\text{c}_\text{a}+9.7\right)}$

$\xi = [\beta (K + \Gamma^*) / (1.6 \eta^*)] ^ {1/2}$

where:

ratio of water lost to carbon fixed [–]
$\text{vpd}$ vapour pressure deficit [Pa]
$\text{c}_\text{a}$ ambient CO2 partial pressure [Pa]
$\chi$ ratio of leaf-internal to ambient CO2 partial pressures [–]
$\xi$ stomatal sensitivity factor [Pa1/2]
$\Gamma^*$ photorespiratory compensation point [Pa]: a function of temperature and elevation
$\beta$ ratio of cost factors for carboxylation and transpiration = 146 [–]
effective Michaelis constant of Rubisco [Pa]: a function of temperature and elevation
$\eta^*$ viscosity of water relative to its value at 25°C [–]

And the equilibrium relation:

$f(\text{T}_\text{c1}, \text{MI}_\text{1}, \text{c}_\text{a,1}) = f(\text{T}_\text{c0}, \text{MI}_\text{0}, \text{c}_\text{a,0})$

where:

$\text{T}_\text{c1}$ past temperature (assume equal to reconstructed value) [K]
$\text{MI}_\text{1}$ past MI (unknown) [–]
$\text{c}_\text{a,1}$ past ambient CO2 partial pressure [Pa], adjusted for elevation
$\text{T}_\text{c0}$ present temperature [K]
$\text{MI}_\text{0}$ reconstructed MI [–]
$\text{c}_\text{a,1}$ ‘recent’ ambient CO2 partial pressure [Pa], adjusted for elevation

Steps in the solution:

Evaluate $f(\text{T}_\text{c0}, \text{MI}_\text{0}, \text{c}_\text{a,0})$
Equate this to:

$[\xi(\text{T}_\text{c1}, z) \times \text{vpd}_1 ^ {1/2} + \text{vpd}_1] / [\text{c}_\text{a,1}(z) / (\text{c}_\text{a,1}(z) + 9.7)]$

where:

is elevation
$\text{vpd}_1$ is past vapour pressure deficit

And solve for $\text{vpd}_1$ .

Convert $\text{vpd}_1$ back to MI (at temperature $\text{T}_\text{c1}$ ), to yield an estimate of $\text{MI}_\text{1}$ .

Using codos, all the steps translate to a simple function call

corrected_mi <- codos::corrected_mi(present_t,
                                    past_temp,
                                    recon_mi,
                                    modern_co2,
                                    past_co2)

Note that this function takes temperatures in [°C] and ambient CO partial pressures in [ $\mu\text{mol}/\text{mol}$ ] (unless, scale_factor is overwritten, e.g. scale_factor = 1 to use ambient CO partial pressures in [Pa]).

More details:

?codos::corrected_mi

References

University of East Anglia Climatic Research Unit, Ian C. Harris, Philip D. Jones, and Tim Osborn. 2020. CRU TS4.04: Climatic Research Unit (CRU) Time-Series (TS) Version 4.04 of High-Resolution Gridded Data of Month-by-Month Variation in Climate (Jan. 1901- Dec. 2019). Centre for Environmental Data Analysis. https://catalogue.ceda.ac.uk/uuid/89e1e34ec3554dc98594a5732622bce9.

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README-extended.md

README-extended.md

CO_dos: CO₂ Correction Tools

Installation

Background:

Vapour-pressure deficit (`vpd`)

Corrected `mi` from reconstructed `mi`

References

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README-extended.md

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README-extended.md

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COdos: CO2 Correction Tools

Installation

Background:

Vapour-pressure deficit (vpd)

Corrected mi from reconstructed mi

References

CO_dos: CO₂ Correction Tools

Vapour-pressure deficit (`vpd`)

Corrected `mi` from reconstructed `mi`