Last updated: 2021-02-20

Checks: 7 0

Knit directory: BloomSail/

This reproducible R Markdown analysis was created with workflowr (version 1.6.2). The Checks tab describes the reproducibility checks that were applied when the results were created. The Past versions tab lists the development history.


Great! Since the R Markdown file has been committed to the Git repository, you know the exact version of the code that produced these results.

Great job! The global environment was empty. Objects defined in the global environment can affect the analysis in your R Markdown file in unknown ways. For reproduciblity it’s best to always run the code in an empty environment.

The command set.seed(20191021) was run prior to running the code in the R Markdown file. Setting a seed ensures that any results that rely on randomness, e.g. subsampling or permutations, are reproducible.

Great job! Recording the operating system, R version, and package versions is critical for reproducibility.

Nice! There were no cached chunks for this analysis, so you can be confident that you successfully produced the results during this run.

Great job! Using relative paths to the files within your workflowr project makes it easier to run your code on other machines.

Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility.

The results in this page were generated with repository version 760a06a. See the Past versions tab to see a history of the changes made to the R Markdown and HTML files.

Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use wflow_publish or wflow_git_commit). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated:


Ignored files:
    Ignored:    .Rhistory
    Ignored:    .Rproj.user/
    Ignored:    data/
    Ignored:    output/Plots/Figures_publication/.tmp.drivedownload/

Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.


These are the previous versions of the repository in which changes were made to the R Markdown (analysis/response_time.Rmd) and HTML (docs/response_time.html) files. If you’ve configured a remote Git repository (see ?wflow_git_remote), click on the hyperlinks in the table below to view the files as they were in that past version.

File Version Author Date Message
Rmd 760a06a jens-daniel-mueller 2021-02-20 cleaning
html 5f4fb9a jens-daniel-mueller 2021-02-20 Build site.
Rmd 031f46a jens-daniel-mueller 2021-02-20 rerun with early exclusion of negative pCO2
html 516b294 jens-daniel-mueller 2021-02-18 Build site.
html 70a8950 jens-daniel-mueller 2021-02-11 Build site.
html 7d39e0b jens-daniel-mueller 2021-02-11 Build site.
Rmd e9b363c jens-daniel-mueller 2021-02-11 cleaning
html acba234 jens-daniel-mueller 2021-02-11 Build site.
Rmd 6d46720 jens-daniel-mueller 2021-02-11 cleaning
html 8051798 jens-daniel-mueller 2021-02-10 Build site.
html 37903c9 jens-daniel-mueller 2021-02-10 Build site.
Rmd 0669cbe jens-daniel-mueller 2021-02-10 italic p in pCO2
html 88b4406 jens-daniel-mueller 2021-01-30 Build site.
Rmd d1fe3c6 jens-daniel-mueller 2021-01-30 resized figures
html 289b763 jens-daniel-mueller 2021-01-30 Build site.
Rmd b470c95 jens-daniel-mueller 2021-01-30 resized figures
html c5fc34c jens-daniel-mueller 2021-01-22 Build site.
Rmd f656a73 jens-daniel-mueller 2021-01-22 all figs revised
html a7950fd jens-daniel-mueller 2021-01-22 Build site.
Rmd 88fcb00 jens-daniel-mueller 2021-01-22 modified figs
html cde4f1d jens-daniel-mueller 2021-01-08 Build site.
Rmd 7cd025c jens-daniel-mueller 2021-01-08 modified figs
html 4277235 jens-daniel-mueller 2021-01-05 Build site.
Rmd 58c1637 jens-daniel-mueller 2021-01-05 new Fig_AX names, A5 added
html 7e29c30 jens-daniel-mueller 2020-11-02 Build site.
Rmd 7e5a700 jens-daniel-mueller 2020-11-02 renamed and revised figures for publication
html 9a3f42a jens-daniel-mueller 2020-10-24 Build site.
html 05248bf jens-daniel-mueller 2020-10-20 Build site.
html 1c4fe8e jens-daniel-mueller 2020-10-20 table with time series in depth intervals added
html 331375c jens-daniel-mueller 2020-10-13 Build site.
Rmd b7d8527 jens-daniel-mueller 2020-10-13 Profiling speed calculated
html 7e3d264 jens-daniel-mueller 2020-10-13 Build site.
Rmd 0c74196 jens-daniel-mueller 2020-10-13 calculated mean pCO2 offset
html 1a55550 jens-daniel-mueller 2020-10-13 Build site.
Rmd bdf7a94 jens-daniel-mueller 2020-10-13 define RT stats criteria in parameterization
html 6896725 jens-daniel-mueller 2020-10-01 Build site.
html 9f66019 jens-daniel-mueller 2020-10-01 Build site.
html 27c5431 jens-daniel-mueller 2020-09-29 Build site.
Rmd 2e0f902 jens-daniel-mueller 2020-09-29 all parameters separate, rebuild
html 1d01685 jens-daniel-mueller 2020-09-28 Build site.
Rmd d28129f jens-daniel-mueller 2020-09-28 republish after tau factor set to 1 and using final pCO2 data
html 1278900 jens-daniel-mueller 2020-09-25 Build site.
html 904f0f7 jens-daniel-mueller 2020-09-23 Build site.
Rmd 7f497e4 jens-daniel-mueller 2020-09-23 updated tau lm fit procedure
html c919fb7 jens-daniel-mueller 2020-06-29 Build site.
Rmd 1461cb6 jens-daniel-mueller 2020-06-29 Fig update for talk
html 603af23 jens-daniel-mueller 2020-05-25 Build site.
html 3414c23 jens-daniel-mueller 2020-05-25 Build site.
html dd3bd89 jens-daniel-mueller 2020-05-07 Build site.
Rmd ad98da2 jens-daniel-mueller 2020-05-07 harmonized parameter labeling
html 113b326 jens-daniel-mueller 2020-05-06 Build site.
Rmd 55b65ca jens-daniel-mueller 2020-05-06 changed example profile
html 6a78488 jens-daniel-mueller 2020-05-06 Build site.
Rmd 5cc31b0 jens-daniel-mueller 2020-05-06 revised selection criteria for summary analytics up vs down
html 907b163 jens-daniel-mueller 2020-05-06 Build site.
Rmd a12b579 jens-daniel-mueller 2020-05-06 unnecessary code and variables removed
html c9ad77d jens-daniel-mueller 2020-05-06 Build site.
Rmd 53baed8 jens-daniel-mueller 2020-05-06 tau high res optimum
html 324117a jens-daniel-mueller 2020-05-05 Build site.
Rmd aa283ad jens-daniel-mueller 2020-05-05 tested 5m grid intervals
html 772e588 jens-daniel-mueller 2020-05-04 Build site.
Rmd 2ab39d7 jens-daniel-mueller 2020-05-04 All profiles and timeseries in one plot pdf
html 3832733 jens-daniel-mueller 2020-04-30 Build site.
Rmd 4f4ab08 jens-daniel-mueller 2020-04-30 harmonized code until RT determination
html 1b6480f jens-daniel-mueller 2020-04-30 Build site.
Rmd fe72316 jens-daniel-mueller 2020-04-30 revised variable and object names, used temp-dependent tau only, rerun code
html b5722a7 jens-daniel-mueller 2020-04-28 Build site.
html 6fdf22f jens-daniel-mueller 2020-04-27 Build site.
Rmd f73f073 jens-daniel-mueller 2020-04-27 removed Date-ID column causing error
html 472c2b4 jens-daniel-mueller 2020-04-21 Build site.
html f8fcf50 jens-daniel-mueller 2020-04-19 created pub figures for time series
html a6c4c22 jens-daniel-mueller 2020-03-30 Build site.
html 80c78b3 jens-daniel-mueller 2020-03-30 Build site.
html 5f8ca30 jens-daniel-mueller 2020-03-20 Build site.
html 2a20453 jens-daniel-mueller 2020-03-20 Build site.
html 473ab25 jens-daniel-mueller 2020-03-19 Build site.
html 81f022e jens-daniel-mueller 2020-03-18 Build site.
html 1e39d85 jens-daniel-mueller 2020-03-18 Build site.
html 2105236 jens-daniel-mueller 2020-03-18 Build site.
html 05b9bdc jens-daniel-mueller 2020-03-17 Build site.
html 0202742 jens-daniel-mueller 2020-03-16 Build site.
html 8e83afd jens-daniel-mueller 2020-03-12 Build site.
html a3ddea4 jens-daniel-mueller 2020-03-12 Build site.
html 52621ea jens-daniel-mueller 2020-03-12 Build site.
html e43a6f2 jens-daniel-mueller 2019-12-19 Build site.
html 3042ff3 jens-daniel-mueller 2019-12-19 Build site.
Rmd 282c3ac jens-daniel-mueller 2019-12-19 whole data set RT corrected
html 0bbafef jens-daniel-mueller 2019-12-13 Build site.
Rmd b9c2b8f jens-daniel-mueller 2019-12-13 Finalized RT correction
html b9c2b8f jens-daniel-mueller 2019-12-13 Finalized RT correction
html 9a4e5c3 jens-daniel-mueller 2019-12-13 Build site.
Rmd cb16002 jens-daniel-mueller 2019-12-13 SD of RT check up vs downcast
html cb16002 jens-daniel-mueller 2019-12-13 SD of RT check up vs downcast
html 5e3fcfb jens-daniel-mueller 2019-12-12 Build site.
Rmd f1cd72d jens-daniel-mueller 2019-12-12 High res tau and RT evaluation
html f1cd72d jens-daniel-mueller 2019-12-12 High res tau and RT evaluation
html 8abe41d jens-daniel-mueller 2019-12-10 Build site.
Rmd 0fcfdbf jens-daniel-mueller 2019-12-10 excluded partial profiles from summary stats
html 70e07ef jens-daniel-mueller 2019-12-10 Build site.
Rmd 935bc85 jens-daniel-mueller 2019-12-10 dep interval profiles decreased to 1m
html 78710ee jens-daniel-mueller 2019-12-09 Build site.
Rmd c6cfca5 jens-daniel-mueller 2019-12-09 RT correction incl OGB data
html 4d3910a jens-daniel-mueller 2019-12-09 Build site.
Rmd 25aeefd jens-daniel-mueller 2019-12-09 data_base included OGB data
html 0e13786 jens-daniel-mueller 2019-12-03 Build site.
Rmd 8da790d jens-daniel-mueller 2019-12-03 tau optimum included
html 4f1a268 jens-daniel-mueller 2019-12-03 Build site.
Rmd 216ad8a jens-daniel-mueller 2019-12-03 response time correction extended
html 216ad8a jens-daniel-mueller 2019-12-03 response time correction extended
html fd0f33d jens-daniel-mueller 2019-11-26 Build site.
Rmd 421c48c jens-daniel-mueller 2019-11-26 found temperature dependence of RT
html 421c48c jens-daniel-mueller 2019-11-26 found temperature dependence of RT
html bc6f19b jens-daniel-mueller 2019-11-22 Build site.
Rmd 03b1b97 jens-daniel-mueller 2019-11-22 updated RT determination
html d921065 jens-daniel-mueller 2019-11-14 Build site.
Rmd 252f84d jens-daniel-mueller 2019-11-14 included EDA in data base
html d61a468 jens-daniel-mueller 2019-11-14 Build site.
Rmd 76e38c6 jens-daniel-mueller 2019-11-14 rebuild site, new toc depth
html 08b9a38 jens-daniel-mueller 2019-11-08 Build site.
Rmd ad4aa12 jens-daniel-mueller 2019-11-08 Finalized RT determination
html ad4aa12 jens-daniel-mueller 2019-11-08 Finalized RT determination
html b8dac9c jens-daniel-mueller 2019-11-08 Build site.
Rmd efcf0d6 jens-daniel-mueller 2019-11-08 Finalized and cleaned RT determination
html f3277a5 jens-daniel-mueller 2019-11-08 Build site.
Rmd aa27fd4 jens-daniel-mueller 2019-11-08 Finalized RT determination
html 4256bcf jens-daniel-mueller 2019-11-08 Build site.
Rmd 7f52e66 jens-daniel-mueller 2019-11-08 response_time pdf updated
html 72687ee jens-daniel-mueller 2019-11-08 Build site.
Rmd 74632d6 jens-daniel-mueller 2019-11-08 response_time pdf updated
html 74212a6 jens-daniel-mueller 2019-11-08 Build site.
Rmd 6cb1935 jens-daniel-mueller 2019-11-08 response_time updated
html 33e3659 jens-daniel-mueller 2019-10-22 Build site.
Rmd efcafd1 jens-daniel-mueller 2019-10-22 Added data base, merging, and RT determination
html 1595fe9 jens-daniel-mueller 2019-10-21 Build site.
html a059c41 jens-daniel-mueller 2019-10-21 Build site.
Rmd eff54ce jens-daniel-mueller 2019-10-21 Added CTD read-in
html 32ec4f7 jens-daniel-mueller 2019-10-21 Build site.
Rmd b2d2bbb jens-daniel-mueller 2019-10-21 Structured data base and response time Rmd
html bafa88f jens-daniel-mueller 2019-10-21 Build site.
Rmd 53ad162 jens-daniel-mueller 2019-10-21 Structured data base and response time Rmd
html 076a36b jens-daniel-mueller 2019-10-21 Build site.
Rmd 3e8a32e jens-daniel-mueller 2019-10-21 Structured data base and response time Rmd
html b2d0164 jens-daniel-mueller 2019-10-21 Build site.
Rmd 53ae361 jens-daniel-mueller 2019-10-21 Added data base and response time Rmd

library(tidyverse)
library(seacarb)
library(broom)
library(lubridate)
library(tibbletime)
library(patchwork)

1 Scope of this script

  • Determine response time \(\tau\) from flush periods of the pCO2 sensor

  • Analyse determined \(\tau\)

  • Apply response time correction to pCO2 data

    • First only to profiles for quality assessment

    • Best parameterization than applied to all data

2 List of relevant parameters

The following aspects were tested and adjusted to improve the performance of the response time correction.

Response time determination

  • Fit interval length
  • tau residual threshold
  • Mean vs T-dependent tau

Response time correction

  • smoothing

Quality assessment of response time correction

  • Difference between up and downcast; comparison to reference value
  • Depth interval width for offset calculation
  • Max depth for Up-down-difference
  • NA criterion for included Down-Up-difference

3 Response time (\(\tau\)) determination

3.1 HydroC sensor settings

The sensor was first run with a low power pump (1W). Later - and for most parts of the expedition - with a stronger (8W) pump. Pumps were switched between recordings (data file: SD_datafile_20180718_170417CO2-0618-001.txt):

  • 2018-07-17;13:08:34
  • 2018-07-17;13:08:35

Logging frequency for all measurement modes (Measure, Zero, Flush) was increased in two steps, It was:

10 sec for all recordings including SD_datafile_20180714_073641CO2-0618-001.txt

2 sec after change in SD_datafile_20180717_121052CO2-0618-001.txt at:

  • 2018-07-14;07:52:02
  • 2018-07-14;07:52:12
  • 2018-07-14;07:52:14

1 sec after change in SD_datafile_20180718_170417CO2-0618-001 at:

  • 2018-07-17;12:27:25
  • 2018-07-17;12:27:27
  • 2018-07-17;12:27:28

3.2 Model fitting

Response times were determined by fitting a nonlinear least-squares model with the nls function as described here by Douglas Watson.

  • Flush period length: variable
  • Flush period restricted to equilibration phase, avoiding initial gas mixing effects occurring at the start of each Flush period
  • only completed Flush periods (duration > 500 sec) included
# read merged data file
tm <-
  read_csv(
    here::here(
      "data/intermediate/_merged_data_files/merging_interpolation",
      "tm.csv"
    ),
    col_types = cols(
      ID = col_character(),
      pCO2_analog = col_double(),
      pCO2_corr = col_double(),
      Zero = col_factor(),
      Flush = col_factor(),
      Zero_counter = col_integer(),
      deployment = col_integer(),
      duration = col_double(),
      mixing = col_character(),
      lat = col_double(),
      lon = col_double()
    )
  )

# select relevant columns
tm <- tm %>%
  select(date_time,
         ID,
         dep,
         tem,
         Flush,
         pCO2_corr,
         Zero_counter,
         duration,
         mixing)

# subset flush data after completed mixing phase
tm_flush <- tm %>%
  filter(Flush == 1, mixing == "equilibration")

# calculate flush duration
tm_flush <- tm_flush %>%
  group_by(Zero_counter) %>%
  mutate(duration = duration - min(duration)) %>%
  ungroup()

rm(tm)

3.2.1 Example plot

An example plot for a nls model fitted to pCO2 observations during a Flush phase is shown below.

# select example flush period
i_Zero_counter <- 51

# set example duration
tm_flush_counter <- tm_flush %>%
  filter(Zero_counter == i_Zero_counter,
         duration <= 300)

# fit RT model
fit <- tm_flush_counter %>%
  nls(pCO2_corr ~ SSasymp(duration, yf, y0, log_alpha), data = .)

# extract relevant model parameters
tau <- as.numeric(exp(-tidy(fit)[3, 2]))
pCO2_corr_end <- as.numeric(tidy(fit)[1, 2])
pCO2_corr_start <- as.numeric(tidy(fit)[2, 2])
pCO2_corr_delta = pCO2_corr_end - pCO2_corr_start
resid_abs_mean <- mean(abs(resid(fit)))

# plot RT fit
augment(fit) %>%
  ggplot(aes(duration, pCO2_corr)) +
  geom_vline(xintercept = tau) +
  geom_hline(yintercept = pCO2_corr_start + 0.63 * (pCO2_corr_delta)) +
  geom_point(shape = 21) +
  geom_line(aes(y = .fitted), col = "red") +
  labs(y = expression(italic(p)*CO[2] ~ (µatm)), x = "time (sec)")
Example response time determination by non-linear least squares fit to the pCO~2~ recovery signal after zeroing. The vertical line indicates the determined response time tau. The horizontal line indicates 63% of the difference between start and final fitted pCO~2~.

Example response time determination by non-linear least squares fit to the pCO2 recovery signal after zeroing. The vertical line indicates the determined response time tau. The horizontal line indicates 63% of the difference between start and final fitted pCO2.

ggsave(
  here::here(
    "output/Plots/Figures_publication/appendix",
    "Fig_A1.pdf"
  ),
  width = 83,
  height = 50,
  dpi = 300,
  units = "mm"
)

ggsave(
  here::here(
    "output/Plots/Figures_publication/appendix",
    "Fig_A1.png"
  ),
  width = 83,
  height = 50,
  dpi = 300,
  units = "mm"
)

rm(
  tm_flush_counter,
  fit,
  i_Zero_counter,
  tau,
  pCO2_corr_delta,
  pCO2_corr_end,
  pCO2_corr_start,
  resid_abs_mean
)

3.2.2 Flush intervals

Due to speculations about the dependence of determined response times (\(\tau\)) on the chosen duration of the fit interval, the response time \(\tau\) was determined for all zeroings and for folllowing duration limits:

150, 200, 250, 300, 350, 400, 450, 500 secs

3.2.3 Fitting algorithm

The code chunk below, fits the response to all Flush periods and duration limits, and creates a pdf with a plot for each individual fit.

pdf(
  file = here::here(
    "output/Plots/response_time",
    "tau_determination_pCO2_corr_flushperiods_nls.pdf"
  ),
  onefile = TRUE,
  width = 7,
  height = 4
)

for (i in unique(tm_flush$Zero_counter)) {
  for (max_duration in parameters$duration_intervals) {
    tm_flush_counter <- tm_flush %>%
      filter(Zero_counter == i, duration <= max_duration)
    
    fit <-
      try(tm_flush_counter %>%
            nls(pCO2_corr ~ SSasymp(duration, yf, y0, log_alpha), data = .),
          TRUE)
    
    if (class(fit) == "nls") {
      tau <- as.numeric(exp(-tidy(fit)[3, 2]))
      pCO2_corr_end <- as.numeric(tidy(fit)[1, 2])
      pCO2_corr_start <- as.numeric(tidy(fit)[2, 2])
      pCO2_corr_delta = pCO2_corr_end - pCO2_corr_start
      resid_abs_mean <- mean(abs(resid(fit)) / pCO2_corr_end) * 100
      
      temp <- as_tibble(
        bind_cols(
          Zero_counter = i,
          duration = max_duration,
          date_time = mean(tm_flush_counter$date_time),
          dep = mean(tm_flush_counter$dep),
          tem = mean(tm_flush_counter$tem),
          pCO2_corr = pCO2_corr_end,
          tau = tau,
          resid = resid_abs_mean
        )
      )
      
      if (exists("tau_values")) {
        tau_values <- bind_rows(tau_values, temp)
      }
      else {
        tau_values <- temp
      }
      
      if (resid_abs_mean > parameters$pCO2_resid_lim) {
        warn <- "orange"
      }
      else {
        warn <- "black"
      }
      
      print(
        augment(fit) %>%
          ggplot(aes(duration, pCO2_corr)) +
          geom_point(col = warn) +
          geom_line(aes(y = .fitted)) +
          geom_vline(xintercept = tau) +
          geom_hline(yintercept = pCO2_corr_start + 0.63 * (pCO2_corr_delta)) +
          labs(
            y = expression(italic(p)*CO[2] ~ (µatm)),
            x = "Duration of Flush period (s)",
            title = paste(
              "Zero_counter: ",
              i,
              "Tau: ",
              round(tau, 1),
              "Mean absolute residual (%): ",
              round(resid_abs_mean, 2)
            )
          ) +
          xlim(0, 600)
      )
      
    }
    
    
    else {
      temp <- as_tibble(
        bind_cols(
          Zero_counter = i,
          duration = max_duration,
          date_time = mean(tm_flush_counter$date_time),
          dep = mean(tm_flush_counter$dep),
          tem = mean(tm_flush_counter$tem),
          pCO2_corr = pCO2_corr_end,
          tau = NaN,
          resid = NaN
        )
      )
      
      if (exists("tau_values")) {
        tau_values <- bind_rows(tau_values, temp)
      }
      else {
        tau_values <- temp
      }
      
      print(
        tm_flush_counter %>%
          ggplot(aes(duration, pCO2_corr)) +
          geom_point(col = "red") +
          labs(
            y = expression(italic(p)*CO[2] ~ (µatm)),
            x = "Duration of Flush period (s)",
            title = paste("Zero_counter: ", i,
                          "nls model failed")
          ) +
          xlim(0, 600)
      )
      
    }
  }
}

dev.off()

rm(
  tm_flush_counter,
  fit,
  i,
  tau,
  pCO2_corr_delta,
  pCO2_corr_end,
  pCO2_corr_start,
  temp,
  max_duration,
  resid_abs_mean,
  warn
)

tau_values %>%
  write_csv(
    here::here(
      "data/intermediate/_merged_data_files/response_time",
      "tau_values.csv"
    )
  )

rm(tm_flush)

A pdf with plots of all individual response time fits can be accessed here.

In this pdf, response time fits that exceed the residual criterion (Mean absolute residual >1% of final pCO2) are printed in orange. Data from flush periods without succesful fit are printed red.

3.2.4 Residual threshold

A mean absolute residual threshold of >1% of final pCO2 was applied to all determined response times.

# define periods of different pumps used
max_Zero_counter <-
  max(unique(tau_values[tau_values$date_time < parameters$pump_switch, ]$Zero_counter))

tau_values <- tau_values %>%
  mutate(pump_power = if_else(Zero_counter <= max_Zero_counter, "1W", "8W"))


# subset determined tau values by residual threshold
tau_resid <- tau_values %>%
  group_by(Zero_counter) %>%
  mutate(resid_max = max(resid, na.rm = TRUE)) %>%
  filter(resid_max < parameters$pCO2_resid_lim) %>%
  select(-resid_max) %>%
  ungroup()

tau_resid_out <- tau_values %>%
  group_by(Zero_counter) %>%
  mutate(resid_max = max(resid, na.rm = TRUE)) %>%
  filter(resid_max > parameters$pCO2_resid_lim) %>%
  select(-resid_max) %>%
  ungroup()

# Flush periods where model failure occurred
tau_values %>%
  filter(is.na(resid)) %>%
  group_by(Zero_counter) %>%
  summarise(n()) %>%
  ungroup()

# Flush periods removed due to residual criterion
tau_resid_out %>%
  group_by(Zero_counter) %>%
  summarise(n()) %>%
  ungroup()

The first determined tau value, which is twice as high as the mean of all others for an unkown reason, was removed.

# mean tau of first RT determination
tau_resid %>%
  filter(Zero_counter == 2) %>%
  summarise(tau = mean(tau))

# mean tau of all RT determinations before pump switch, except first
tau_resid %>%
  filter(Zero_counter != 2, Zero_counter <= 20) %>%
  summarise(tau = mean(tau))

# remove first tau value which is twice as high as the mean of all others
tau_resid <- tau_resid %>%
  filter(Zero_counter != 2)

3.3 Analysis

3.3.1 General considerations

Estimated \(\tau\) values were only taken into account when stable environmental pCO2 levels were present. Absence of stable environmental pCO2 was assumed when the mean absolute fit residual was above 1 % of the final equilibrium pCO2. If one model fit (irrespective the chosen fit interval length) of a particular flush period did not match that criterion, the flush period was ignored entirely. Usually, fits with higher duration did not meet this criterion. For some unexplained reason the first \(\tau\) determination resulted in values about twice as high as all other flush periods and was therefore removed as an outlier.

Metrics to characterize the fitting procedure include the number of:

n_Zero_counters <- tau_values %>%
  group_by(Zero_counter) %>%
  n_groups()

n_duration_intervals <-
  length(parameters$duration_intervals)

n_tau_max <- n_Zero_counters * length(parameters$duration_intervals)
n_tau_total <- nrow(tau_values %>% filter(!is.na(resid)))
n_tau_resid <- nrow(tau_resid)
  • Flush periods: 76
  • Duration intervals: 8
  • Exercised response time fits: 608
  • Succesful response times determinations: 576 (94.7)%
  • \(\tau\)’s after removing groups of fits with high absolute fit residual: 416 (68.4 %)

It should be noted that all failed model fits occured in flush periods where the residual criterion was not meet by at least one other fit (i.e. fitting only failed under unstable conditions).

tau_values %>%
  ggplot(aes(resid)) +
  geom_histogram() +
  facet_wrap( ~ duration, labeller = label_both) +
  geom_vline(xintercept = parameters$pCO2_resid_lim) +
  labs(x = expression(Mean ~ absolute ~ residuals ~ ("%" ~ of ~ equilibrium ~
                                                       italic(p)*CO[2])))
Histogram of residuals from fit displayed for the investigate durations of the fit interval. Vertical line represents the chosen threshold.

Histogram of residuals from fit displayed for the investigate durations of the fit interval. Vertical line represents the chosen threshold.

3.3.2 Fit interval length

No clear dependence of \(\tau\) on the length of the flushing period was found.

tau_resid %>%
  group_by(Zero_counter) %>%
  mutate(d_tau = tau - mean(tau)) %>%
  ggplot(aes(duration, d_tau)) +
  geom_hline(yintercept = 0) +
  geom_smooth() +
  geom_point() +
  facet_wrap( ~ Zero_counter, ncol = 4, labeller = label_both) +
  labs(x = "Duration (sec)", y = "Deviation from mean tau (sec)")
Determined tau values as a function of the fit interval duration, displayed individually for each flush period.

Determined tau values as a function of the fit interval duration, displayed individually for each flush period.

tau_resid %>%
  group_by(Zero_counter) %>%
  mutate(d_tau = tau - mean(tau)) %>%
  ggplot(aes(duration, d_tau, group = duration)) +
  geom_violin() +
  geom_point() +
  labs(x = "Duration (sec)", y = "Deviation from mean tau (sec)") +
  facet_wrap( ~ pump_power)
Determined tau values as a function of the fit interval duration, pooled for all flush period.

Determined tau values as a function of the fit interval duration, pooled for all flush period.

duration_min_tau_sd <- tau_resid %>%
  group_by(Zero_counter) %>%
  mutate(d_tau = tau - mean(tau)) %>%
  ungroup() %>%
  group_by(duration) %>%
  summarise(d_tau_sd = sd(d_tau, na.rm = TRUE)) %>%
  ungroup() %>%
  slice(which.min(d_tau_sd)) %>%
  select(duration) %>%
  pull()

The lowest standard deviation of \(\tau\) values was found for a duration of:

  • 300 sec

\(\tau\) values determined with this duration were filtered for further analysis.

tau_resid <- tau_resid %>%
  filter(duration == duration_min_tau_sd) %>%
  select(-duration)

rm(duration_min_tau_sd) 

3.3.3 Time series of \(\tau\)

No obvious change of \(\tau\) over time was detected, but a dependence on the pump used.

ggplot() +
  geom_smooth(
    data = tau_resid %>% filter(dep < 10),
    aes(date_time, tau, linetype = pump_power),
    method = "lm",
    se = FALSE,
    col = "red"
  ) +
  geom_point(data = tau_resid,
             aes(date_time, tau, col = dep, shape = pump_power)) +
  scale_color_viridis_c(name = "Depth (m)") +
  labs(y = "Tau (sec)", x = "Date") +
  ylim(0, NA)
Tau for all Zeroings with color representing water depth. Red lines represent linear regression trends for tau determined in surface waters (<10m).

Tau for all Zeroings with color representing water depth. Red lines represent linear regression trends for tau determined in surface waters (<10m).

3.3.4 Temperature dependence

A temperature dependence of determined response times \(\tau\) was found, with similar slopes but different intercepts for both pumps used.

tau_resid %>%
  ggplot(aes(tem, tau, col = dep)) +
  geom_smooth(method = "lm") +
  geom_point() +
  scale_color_viridis_c(name = "Depth (m)") +
  labs(y = "Tau (sec)", x = "Temperature (deg C)") +
  facet_wrap( ~ pump_power, labeller = label_both)
Tau as a function of temperature for all zeroings determined with low power (left) and strong (right) pump. Color represents the water depth.

Tau as a function of temperature for all zeroings determined with low power (left) and strong (right) pump. Color represents the water depth.

For the response times determined near the surface (<10m, restricted temperature range), no clear temperature dependence of \(\tau\) was detected.

tau_resid %>%
  filter(dep < 10) %>%
  ggplot(aes(tem, tau, col = dep)) +
  geom_smooth(method = "lm") +
  geom_point() +
  scale_color_viridis_c(name = "Depth (m)") +
  labs(y = "Tau (sec)", x = "Temperature (deg C)") +
  facet_wrap( ~ pump_power, labeller = label_both)
Surface tau (<10m) as a function of temperature for all zeroings determined with low power (left) and strong (right) pump. Color represents the water depth.

Surface tau (<10m) as a function of temperature for all zeroings determined with low power (left) and strong (right) pump. Color represents the water depth.

3.3.5 Final \(\tau\) values

The mean \(\tau\) values are:

# calculate mean tau for each pump
tau_mean <- tau_resid %>% 
  group_by(pump_power) %>% 
  summarise(tau = mean(tau, na.rm = TRUE))

tau_mean
# A tibble: 2 x 2
  pump_power   tau
  <chr>      <dbl>
1 1W          76.7
2 8W          56.5

The linear response of \(\tau\) on water temperature was fitted as:

# fit linear regression of tau for each pump
tau_fit <- tau_resid %>%
  nest_by(pump_power) %>%
  mutate(fit = list(lm(tau ~ tem, data = data))) %>%
  summarise(tidy(fit)) %>%
  select(pump_power, term, estimate) %>%
  spread(term, estimate)

tau_fit
# A tibble: 2 x 3
# Groups:   pump_power [2]
  pump_power `(Intercept)`    tem
  <chr>              <dbl>  <dbl>
1 1W                  94.5 -1.12 
2 8W                  70.4 -0.648
tau_fit %>% write_csv(here::here(
  "data/intermediate/_merged_data_files/response_time",
  "tau_fit.csv"
))

# clean workspace
rm(list = setdiff(ls(), c(
  "tau_resid", "tau_fit", "parameters"
)))

Only the T-dependent \(\tau\) estimate will be applied to correct the recorded pCO2 profiles.

4 Response time correction

4.1 Data preparation

Following tasks were performed to prepare data for the response time correction:

  • Select only profiles
  • Assign deployment periods with 1W- and 8W- pump
tm <- read_csv(here::here("data/intermediate/_merged_data_files/merging_interpolation",
                          "tm.csv"),
               col_types = cols(ID = col_character(),
                                pCO2_analog = col_double(),
                                pCO2_corr = col_double(),
                                Zero = col_factor(),
                                Flush = col_factor(),
                                Zero_counter = col_integer(),
                                deployment = col_integer(),
                                duration = col_double(),
                                mixing = col_character(),
                                lat = col_double(),
                                lon = col_double()))

# select relevant columns
tm <- tm %>%
  select(date_time, ID, type, station, dep, sal, tem,
         Zero, Flush, pCO2_corr, deployment, Zero_counter)

# filter profiles
tm <- tm %>% 
  filter(type == "P")

# assign pump types
tm <- tm %>%
  group_by(ID, station) %>% 
  mutate(duration = as.numeric(date_time - min(date_time)),
         pump_power = if_else(date_time < parameters$pump_switch, "1W", "8W")) %>%
  arrange(date_time)
  • Include manually derived meta-information about the profiling status. Those meta data were assigned through visual inspection of each profiles (depth vs time) and distinguish phases of the casts, in particular the continuous up and downcast operation.
# Load profile meta data 
meta <- read_csv(here::here("data/input/TinaV/Sensor",
                            "Sensor_meta.csv"),
                 col_types = cols(ID = col_character()))


# Merge profiles and meta information 
tm <- full_join(tm, meta)
rm(meta)


# assign profiling phases according to meta data

tm <- tm %>% 
  mutate(phase = "standby",
         phase = if_else(duration >= start &
                           duration < down & 
                           !is.na(down) &
                           !is.na(start),
                         "down", phase),
         phase = if_else(duration >= down  & 
                           duration < lift & 
                           !is.na(lift) & 
                           !is.na(down ),   
                         "low",  phase),
         phase = if_else(duration >= lift  & 
                           duration < up   & 
                           !is.na(up  ) & 
                           !is.na(lift  ),  
                         "mid",  phase),
         phase = if_else(duration >= up    & 
                           duration < end  & 
                           !is.na(end ) & 
                           !is.na(up   ),   
                         "up",   phase))

tm <- tm %>% 
  select(-c(start, down, lift, up, end, comment))

# discard zero and flush periods
tm <- tm %>% 
  filter(Zero == 0, Flush == 0)
  • Subset reference pCO2 recordings at the end of equilibration periods executed at constant depth
tm_pCO2_equi <- tm %>% 
  filter(phase %in% c("mid")) %>% 
  group_by(ID, station) %>% 
  top_n(5, row_number()) %>% 
  summarise(date_time = mean(date_time),
            duration = mean(duration),
            pCO2_corr = mean(pCO2_corr, na.rm = TRUE),
            dep = mean(dep, na.rm = TRUE)) %>% 
  ungroup()


# tm_pCO2_equi %>% 
#  write_csv(here::here("data/intermediate/_merged_data_files/response_time",
#                       "tm_pCO2_equi.csv"))
cast_dep <- tm %>%
  pivot_longer(c(dep, pCO2_corr), names_to = "parameter", values_to = "value")

cast_dep_equi <- tm_pCO2_equi %>%
  pivot_longer(c(dep, pCO2_corr), names_to = "parameter", values_to = "value")

max_duration <- round(max(cast_dep$duration) / 1000, 0) * 1000

cast_dep_equi_sub <- cast_dep_equi %>%
  filter(ID == parameters$example_ID,
         station == parameters$example_station)


cast_dep %>%
  filter(ID == parameters$example_ID,
         station == parameters$example_station) %>%
  ggplot(aes(duration, value, col = phase)) +
  geom_point(size = 0.5) +
  geom_point(data = cast_dep_equi_sub, aes(duration, value), col = "black") +
  scale_y_reverse() +
  scale_x_continuous(breaks = seq(0, 6000, 500)) +
  labs(title = str_c(
    "Date: ",
    parameters$example_ID,
    " | Station: ",
    parameters$example_station
  )) +
  facet_grid(parameter ~ ., scales = "free_y")
Example timeseries of profiling depth and pCO~2~. Colors represent manually assigned profiling phases. The black points represent reference data collected at the end of the mid equilibration period.

Example timeseries of profiling depth and pCO2. Colors represent manually assigned profiling phases. The black points represent reference data collected at the end of the mid equilibration period.

rm(cast_dep, cast_dep_equi, cast_dep_equi_sub, max_duration)
cast_dep <- tm %>% 
  pivot_longer(c(dep, pCO2_corr), names_to = "parameter", values_to = "value")

cast_dep_equi <- tm_pCO2_equi %>% 
  pivot_longer(c(dep, pCO2_corr), names_to = "parameter", values_to = "value")

max_duration <- round(max(cast_dep$duration)/1000,0)*1000

pdf(file=here::here("output/Plots/response_time",
                    "time_series_depth_pCO2_corr_by_profile.pdf"),
    onefile = TRUE, width = 7, height = 4)

for(i_ID in unique(cast_dep$ID)){
  for(i_station in unique(cast_dep$station)){

    if (nrow(cast_dep %>% filter(ID == i_ID, station == i_station)) > 0){
      
        cast_dep_equi_sub <- cast_dep_equi %>%
        filter(ID == i_ID,
               station == i_station)

      print(
      
        cast_dep %>%
        filter(ID == i_ID,
               station == i_station) %>%
        ggplot(aes(duration, value, col=phase))+
        geom_point(size=0.5)+
        geom_point(data = cast_dep_equi_sub, aes(duration, value), col="black")+
        scale_y_reverse()+
        scale_x_continuous(breaks = seq(0,6000,500))+
        labs(title = str_c("Date: ",i_ID," | Station: ",i_station))+
        facet_grid(parameter~., scales = "free_y")+
        theme_bw()
      
      )

    }

  }
}

dev.off()

rm(cast_dep, cast_dep_equi, cast_dep_equi_sub, i_station, i_ID, max_duration)

A pdf with all timeseries plots of profiling depth and pCO2 can be accessed here.

4.1.1 Profiling speed

We plotted a histogram of mean downcast profiling speed (m/s) per profile and calculated the mean profiling speed.

downcast <- tm %>%
  filter(phase == "down")

downcast_speed <- downcast %>%
  group_by(ID, station) %>%
  summarise(
    duration = as.numeric(max(date_time) - min(date_time)),
    distance = max(dep) - min(dep),
    speed = distance / duration
  ) %>%
  ungroup()

downcast_speed %>% 
  ggplot(aes(speed)) +
  geom_histogram()

downcast_speed %>% 
  summarise(mean(speed),
            sd(speed))
# A tibble: 1 x 2
  `mean(speed)` `sd(speed)`
          <dbl>       <dbl>
1          1.95       0.448
rm(downcast, downcast_speed)

4.2 Apply correction

The executed response time correction featured the following aspects:

  • Correction according to Bittig et al. (2018, supplement)
  • RT: T-dependent response times applied
  • tau_factor: Factors of 0.8, 1, 1.2, 1.4, 1.6 were applied to determined tau values
  • Post-smoothing: 30 sec running mean (eg across 15 observations at 2 sec measurement frequency)
# Define function for response time correction after Bittig, 2018

RT_corr <- function(c1, c0, dt, tau) {
  (1 / (2 * ((1 + (
    2 * tau / dt
  )) ^ (-1)))) * (c1 - (1 - (2 * ((
    1 + (2 * tau / dt)
  ) ^ (
    -1
  )))) * c0)
}


# Assign T-dependent response time (tau) values

tau_fit <- tau_fit %>%
  rename(tau_intercept = `(Intercept)`, tau_slope = tem)

tm <- full_join(tm, tau_fit)

tm <- tm %>%
  mutate(tau = tau_intercept + tau_slope * tem) %>%
  select(-tau_intercept,-tau_slope)


# Prepare data set for RT correction

tm <- tm %>%
  arrange(date_time) %>%
  mutate(dt = as.numeric(as.character(date_time - lag(date_time))))


# determine measurement frequency of sensor

freq <- tm %>%
  filter(dt < 13) %>%
  group_by(ID) %>%
  summarise(dt_mean = round(mean(dt, na.rm = TRUE), 0))

tm <- full_join(tm, freq)

# apply tau factors

tm <- expand_grid(tm, tau_factor = parameters$tau_factors)

tm <- tm %>%
  mutate(tau_test = tau * tau_factor)


# Apply RT correction to profiling data

for (i_ID in unique(tm$ID)) {

  freq_sub <- freq %>% filter(ID == i_ID) %>% pull(dt_mean)
  
  # window width for smoothing
  window <- parameters$smoothing_duration / freq_sub
  # function for rolling mean
  rolling_mean   <-
    rollify( ~ mean(.x, na.rm = TRUE), window = window)
  
  # data subset for each cruise day, and RT correction per station
  tm_sub <- tm %>%
    filter(ID == i_ID) %>%
    group_by(station, tau_factor) %>%
    mutate(
      pCO2_RT = RT_corr(pCO2_corr, lag(pCO2_corr), dt, tau_test),
      pCO2_RT = if_else(pCO2_RT %in% c(Inf,-Inf), NaN, pCO2_RT),
      window = window,
      pCO2 = rolling_mean(pCO2_RT)
    ) %>%
    ungroup()
  
  # time shift RT corrected data
  shift <- as.integer(as.character(window / 2))
  
  tm_sub <- tm_sub %>%
    group_by(station, tau_factor) %>%
    mutate(pCO2 = lead(pCO2, shift)) %>%
    ungroup()
  
  # append to new data frame
  if (exists("tm_RT")) {
    tm_RT <- bind_rows(tm_RT, tm_sub)
  }
  else{
    tm_RT <- tm_sub
  }
  
  rm(tm_sub, freq_sub, rolling_mean, shift, window)
  
}

tm <- tm_RT

rm(i_ID, freq, RT_corr, tau_fit, tau_resid, tm_RT)

5 Diagnosis

In the following, the success of the response time correction is assessed through the

  • visual inspection of RT corrected profiles,

as well as the offset between the downcast and:

  • Upcast
  • pCO2 reference value (recorded after equilibration period during upcast)

The offset comparison requires to discretize the continous depth recording. Depth intervals of 1m were chosen.

First, we analyze all profiles individually. Later we’ll merge the information across profiles and come up with a single metric to quantify the quality of the response time correction

5.1 Individual profiles

equi_cast <- tm_pCO2_equi %>%
  filter(ID == parameters$example_ID,
         station == parameters$example_station)

tm %>%
  filter(
    ID == parameters$example_ID,
    station == parameters$example_station,
    phase %in% c("up", "down")
  ) %>%
  ggplot() +
  geom_path(aes(pCO2_corr, dep, linetype = phase, col = "raw")) +
  geom_path(aes(pCO2, dep, linetype = phase, col = "corrected")) +
  geom_point(data = equi_cast, aes(pCO2_corr, dep)) +
  scale_y_reverse() +
  coord_cartesian(ylim = c(35, 0),
                  xlim = c(70, 270)) +
  scale_color_brewer(palette = "Set1", name = "") +
  labs(title = str_c(
    "Date: ",
    parameters$example_ID,
    " | Station: ",
    parameters$example_station
  )) +
  facet_grid(tau_factor ~ ., labeller = label_both)
Example plot of response time corrected and raw pCO~2~ profiles. Panels highlight the effect of constant vs T-dependent tau estimates (columns) and the optimization by applying a constant factor (rows).

Example plot of response time corrected and raw pCO2 profiles. Panels highlight the effect of constant vs T-dependent tau estimates (columns) and the optimization by applying a constant factor (rows).

rm(equi_cast)

A pdf with all timeseries plots of profiling depth and pCO2 can be accessed here

pdf(
  file = here::here("output/Plots/response_time",
                    "profiles_pCO2.pdf"),
  onefile = TRUE,
  width = 7,
  height = 11
)

for (i_ID in unique(tm$ID)) {
  for (i_station in unique(tm$station)) {
    if (nrow(tm %>% filter(ID == i_ID, station == i_station)) > 0) {
      equi_cast <- tm_pCO2_equi %>%
        filter(ID == i_ID,
               station == i_station)
      
      print(
        tm %>%
          filter(ID == i_ID,
                 station == i_station,
                 phase %in% c("up", "down")) %>%
          ggplot() +
          geom_path(aes(
            pCO2_corr, dep, linetype = phase, col = "raw"
          )) +
          geom_path(aes(
            pCO2, dep, linetype = phase, col = "corrected"
          )) +
          geom_point(data = equi_cast, aes(pCO2_corr, dep)) +
          scale_y_reverse() +
          coord_cartesian(ylim = c(35, 0),
                          xlim = c(70, 270)) +
          scale_color_brewer(palette = "Set1", name = "") +
          labs(title = str_c(
            "Date: ", i_ID, " | Station: ", i_station
          )) +
          theme_bw() +
          facet_grid(tau_factor ~ ., labeller = label_both)
        
      )
      
      
    }
    
  }
}

dev.off()

rm(equi_cast, i_ID, i_station)

5.2 Down- vs upcast

# pCO2 offset up - down cast
tm_grid <- tm %>%
  filter(phase %in% c("down", "up")) %>%
  mutate(dep_grid = as.numeric(as.character(cut(
    dep, seq(0, 40, 1), seq(0.5, 39.5, 1)
  ))),
  tau_factor = as.factor(tau_factor)) %>%
  select(ID, station, tau_factor, p_type, dep_grid, phase, pCO2_corr, pCO2) %>%
  group_by(ID, station, tau_factor, p_type, dep_grid, phase) %>%
  summarise_all("mean", na.rm = TRUE) %>%
  ungroup() %>%
  pivot_longer(cols = c(pCO2_corr, pCO2), names_to = "correction") %>%
  pivot_wider(names_from = phase, values_from = value) %>%
  mutate(
    pCO2_delta = up - down,
    pCO2_up_down_average = (down + up) / 2,
    pCO2_delta_rel = 100 * pCO2_delta / pCO2_up_down_average
  )
tm_pCO2_equi_grid <- tm_pCO2_equi %>%
  mutate(dep_grid = as.numeric(as.character(cut(
    dep, seq(0, 40, 1), seq(0.5, 39.5, 1)
  )))) %>%
  select(ID, station, dep_grid, pCO2_equi = pCO2_corr)
tm_pCO2_equi_grid_sub <- tm_pCO2_equi_grid %>%
  filter(ID == parameters$example_ID,
         station == parameters$example_station)


tm_grid %>%
  filter(ID == parameters$example_ID,
         station == parameters$example_station) %>%
  arrange(dep_grid) %>%
  ggplot() +
  geom_path(aes(down, dep_grid, col = correction, linetype = "down")) +
  geom_path(aes(up, dep_grid, col = correction, linetype = "up")) +
  geom_point(data = tm_pCO2_equi_grid_sub, aes(pCO2_equi, dep_grid)) +
  scale_y_reverse() +
  coord_cartesian(ylim = c(35, 0),
                  xlim = c(70, 270)) +
  scale_linetype(name = "cast") +
  scale_color_brewer(palette = "Set1", direction = -1) +
  labs(
    y = "Depth [m]",
    x = expression(italic(p) * CO[2] ~ (µatm)),
    title = str_c(
      "Date: ",
      parameters$example_ID,
      " | Station: ",
      parameters$example_ID
    )
  ) +
  facet_grid(tau_factor ~ ., labeller = label_both)
Example plot of discretized, response time corrected and raw pCO~2~ profiles. Panels highlight the effect of constant vs T-dependent tau estimates (columns) and the optimization by applying a constant factor (rows). The black point indicates the reference pCO~2~ value.

Example plot of discretized, response time corrected and raw pCO2 profiles. Panels highlight the effect of constant vs T-dependent tau estimates (columns) and the optimization by applying a constant factor (rows). The black point indicates the reference pCO2 value.

rm(tm_pCO2_equi_grid_sub)

A pdf with all discretized pCO2 profiles can be assessed here

pdf(
  file = here::here("output/Plots/response_time",
                    "profiles_pCO2_grid.pdf"),
  onefile = TRUE,
  width = 7,
  height = 11
)

for (i_ID in unique(tm_grid$ID)) {
  for (i_station in unique(tm_grid$station)) {
    if (nrow(tm_grid %>% filter(ID == i_ID, station == i_station)) > 0) {
      tm_pCO2_equi_grid_sub <- tm_pCO2_equi_grid %>%
        filter(ID == i_ID,
               station == i_station)
      
      print(
        tm_grid %>%
          filter(ID == i_ID,
                 station == i_station) %>%
          arrange(dep_grid) %>%
          ggplot() +
          geom_path(aes(
            down, dep_grid, col = correction, linetype = "down"
          )) +
          geom_path(aes(
            up, dep_grid, col = correction, linetype = "up"
          )) +
          geom_point(data = tm_pCO2_equi_grid_sub, aes(pCO2_equi, dep_grid)) +
          scale_y_reverse() +
          coord_cartesian(ylim = c(35, 0),
                          xlim = c(70, 270)) +
          scale_linetype(name = "cast") +
          scale_color_brewer(palette = "Set1", direction = -1) +
          labs(
            y = "Depth [m]",
            title = str_c("Date: ", i_ID, " | Station: ", i_station)
          ) +
          theme_bw() +
          facet_grid(tau_factor ~ ., labeller = label_both)
        
      )
      
      rm(tm_pCO2_equi_grid_sub)
    }
    
  }
}

dev.off()
tm_grid %>%
  filter(
    ID == parameters$example_ID,
    station == parameters$example_station,
    correction == "pCO2"
  ) %>%
  arrange(dep_grid) %>%
  ggplot(aes(pCO2_delta, dep_grid, col = as.factor(tau_factor))) +
  geom_path() +
  geom_point() +
  scale_y_reverse(breaks = seq(0, 40, 2)) +
  scale_color_discrete(name = "tau factor") +
  labs(
    x = expression(Delta ~ italic(p) * CO[2] ~ (µatm)),
    y = "Depth [m]",
    title = str_c(
      "Date: ",
      parameters$example_ID,
      " | Station: ",
      parameters$example_station
    )
  ) +
  geom_vline(xintercept = 0) +
  geom_vline(xintercept = c(-10, 10), col = "red")
Example plot of absolute pCO~2~ offset profiles. Panels highlight the effect of constant vs T-dependent tau estimates. Colour indicates the optimization by applying a constant factor to tau. Vertical red lines mark an arbitray 10 µatm pCO~2~ threshold.

Example plot of absolute pCO2 offset profiles. Panels highlight the effect of constant vs T-dependent tau estimates. Colour indicates the optimization by applying a constant factor to tau. Vertical red lines mark an arbitray 10 µatm pCO2 threshold.

pdf(
  file = here::here(
    "output/Plots/response_time",
    "profiles_pCO2_delta_grid.pdf"
  ),
  onefile = TRUE,
  width = 7,
  height = 7
)

for (i_ID in unique(tm_grid$ID)) {
  for (i_station in unique(tm_grid$station)) {
    if (nrow(tm_grid %>% filter(ID == i_ID, station == i_station)) > 0) {
      print(
        tm_grid %>%
          filter(ID == i_ID,
                 station == i_station,
                 correction == "pCO2") %>%
          arrange(dep_grid) %>%
          ggplot(aes(
            pCO2_delta, dep_grid, col = as.factor(tau_factor)
          )) +
          geom_path() +
          geom_point() +
          scale_y_reverse(breaks = seq(0, 40, 2)) +
          scale_color_discrete(name = "tau factor") +
          labs(
            x = "delta pCO2 [µatm]",
            y = "Depth [m]",
            title = str_c("Date: ", i_ID, " | Station: ", i_station)
          ) +
          geom_vline(xintercept = 0) +
          geom_vline(xintercept = c(-10, 10), col = "red") +
          theme_bw()
        
      )
      
    }
    
  }
}

dev.off()

A pdf with all absolute pCO2 offset profiles can be assessed here.

tm_grid %>%
  filter(
    ID == parameters$example_ID,
    station == parameters$example_station,
    correction == "pCO2"
  ) %>%
  arrange(dep_grid) %>%
  ggplot(aes(pCO2_delta_rel, dep_grid, col = as.factor(tau_factor))) +
  geom_path() +
  geom_point() +
  scale_y_reverse(breaks = seq(0, 40, 2)) +
  scale_color_discrete(name = "tau factor") +
  labs(
    x = expression(Delta ~ italic(p) * CO[2] ~ ("%" ~ of ~ absolute ~ value)),
    y = "Depth [m]",
    title = str_c(
      "Date: ",
      parameters$example_ID,
      " | Station: ",
      parameters$example_station
    )
  ) +
  geom_vline(xintercept = 0) +
  geom_vline(xintercept = c(-10, 10), col = "red")
Example plot of relative offset pCO~2~ profiles. Panels highlight the effect of constant vs T-dependent tau estimates. Colour indicates the optimization by applying a constant factor to tau. Vertical red lines mark an arbitray 10% threshold.

Example plot of relative offset pCO2 profiles. Panels highlight the effect of constant vs T-dependent tau estimates. Colour indicates the optimization by applying a constant factor to tau. Vertical red lines mark an arbitray 10% threshold.

A pdf with all relative pCO2 offset profiles can be assessed here.

pdf(
  file = here::here(
    "output/Plots/response_time",
    "profiles_pCO2_delta_rel_grid.pdf"
  ),
  onefile = TRUE,
  width = 7,
  height = 7
)

for (i_ID in unique(tm_grid$ID)) {
  for (i_station in unique(tm_grid$station)) {
    if (nrow(tm_grid %>% filter(ID == i_ID, station == i_station)) > 0) {
      print(
        tm_grid %>%
          filter(ID == i_ID,
                 station == i_station,
                 correction == "pCO2") %>%
          arrange(dep_grid) %>%
          ggplot(aes(
            pCO2_delta_rel, dep_grid, col = as.factor(tau_factor)
          )) +
          geom_path() +
          geom_point() +
          scale_y_reverse(breaks = seq(0, 40, 2)) +
          scale_color_discrete(name = "tau factor") +
          labs(
            x = "delta pCO2 [% of absolute value]",
            y = "Depth [m]",
            title = str_c("Date: ", i_ID, " | Station: ", i_station)
          ) +
          geom_vline(xintercept = 0) +
          geom_vline(xintercept = c(-10, 10), col = "red") +
          theme_bw()
        
      )
      
    }
    
  }
}

dev.off()

5.3 Downcast vs reference value

Referenced values were obtained occasionally through interuption of the profiling measurements to achieve sensor equilibration. Readings under equilibrated conditions were extracted and compared to the corresponding continous cast value.

tm_equi_delta <- full_join(tm_grid, tm_pCO2_equi_grid) %>% 
  filter(!is.na(pCO2_equi)) %>% 
  mutate(pCO2_delta_equi = down - pCO2_equi,
         pCO2_delta_equi_rel = 100 * pCO2_delta_equi / pCO2_equi)
tm_equi_delta %>%
  filter(tau_factor == parameters$tau_factor_used, correction == "pCO2") %>%
  ggplot(aes(pCO2_equi, pCO2_delta_equi)) +
  geom_hline(yintercept = 0) +
  geom_point() +
  labs(x = expression(Reference ~ italic(p)*CO[2] ~ (µatm)),
       y = expression(Delta ~ italic(p)*CO[2] ~ from ~ reference ~ (µatm)))
Offset between pCO~2~ downcast and upcast reference value as a function of absolute pCO~2~.

Offset between pCO2 downcast and upcast reference value as a function of absolute pCO2.

tm_equi_delta %>%
  ggplot(aes(as.factor(tau_factor), pCO2_delta_equi, fill = correction)) +
  geom_hline(yintercept = 0) +
  geom_violin() +
  scale_fill_brewer(palette = "Set1")
Offset between pCO~2~ downcast and upcast reference value. Panels highlight the effect of constant vs T-dependent tau estimates. Colour distinguish raw and corrected offsets.

Offset between pCO2 downcast and upcast reference value. Panels highlight the effect of constant vs T-dependent tau estimates. Colour distinguish raw and corrected offsets.

5.4 Summary metrics

In order to decide, which conditions resulted in the best response correction the mean absoulte and relative pCO2 offset across all profiles was calculated for:

  • the offset between downcast and reference value
  • the offset between downcast and upcast
  • T-dependent tau
  • applied tau factors
tm_grid_stat <- tm_grid %>% 
  filter(correction == "pCO2") %>% 
  group_by(ID, station) %>% 
  summarise(dep_max = max(dep_grid),
            pCO2_max = max(down)) %>% 
  ungroup()

tm_grid_stat %>% 
  ggplot(aes(dep_max, pCO2_max)) +
  geom_point()

tm_grid <- full_join(tm_grid, tm_grid_stat)

meta <- read_csv(here::here("data/input/TinaV/Sensor",
                            "Sensor_meta.csv"),
                 col_types = cols(ID = col_character()))

tm_grid_stat <- full_join(meta, tm_grid_stat)

rm(tm_grid_stat, meta)

Summary metric are restricted to profiles that did not exceed:

  • maximum sampling depth of 30 m
  • pCO2 of 300 µatm
  • Nr of missing depth grid levels of 2
# apply restrictions
tm_grid_shallow <- tm_grid %>% 
  filter(dep_max <= parameters$RT_stats_dep_max,
         pCO2_max <= parameters$RT_stats_pCO2_max,
         dep_grid <= parameters$RT_stats_dep) %>% 
  group_by(ID, station, tau_factor, correction) %>% 
  mutate(nr_na = sum(is.na(pCO2_delta))) %>% 
  ungroup() %>% 
  filter(nr_na < parameters$max_gap)

# calculate summary metrics up vs downcast
tm_grid_shallow_sum <- tm_grid_shallow %>% 
  mutate(pCO2_delta_abs = abs(pCO2_delta),
         pCO2_delta_rel_abs = abs(pCO2_delta_rel)) %>% 
  group_by(tau_factor, dep_grid, correction) %>% 
  summarise(mean         = mean(pCO2_delta, na.rm = TRUE),
            sd           = sd(pCO2_delta, na.rm = TRUE),
            mean_abs     = mean(pCO2_delta_abs, na.rm = TRUE),
            mean_rel     = mean(pCO2_delta_rel, na.rm = TRUE),
            sd_rel       = sd(pCO2_delta_rel, na.rm = TRUE),
            mean_rel_abs = mean(pCO2_delta_rel_abs, na.rm = TRUE)) %>% 
  ungroup() %>% 
  pivot_longer(cols = sd:mean_rel_abs,
               names_to = "estimate", values_to = "pCO2_delta")

# calculate summary metrics equilibrium comparison
tm_equi_delta_sum <- tm_equi_delta %>%
  mutate(pCO2_delta_equi_abs = abs(pCO2_delta_equi),
         pCO2_delta_equi_rel_abs = abs(pCO2_delta_equi_rel)) %>%
  group_by(correction, tau_factor) %>%
  summarise(mean         = mean(pCO2_delta_equi, na.rm = TRUE),
            mean_abs     = mean(pCO2_delta_equi_abs, na.rm = TRUE),
            mean_rel = mean(pCO2_delta_equi_rel, na.rm = TRUE),
            mean_rel_abs = mean(pCO2_delta_equi_rel_abs, na.rm = TRUE)) %>%
  ungroup() %>%
  pivot_longer(cols = mean:mean_rel_abs,
               names_to = "estimate", values_to = "dpCO2")
tm_grid_shallow_sum %>%
  filter(correction == "pCO2",
         estimate %in% c("mean_abs", "mean_rel_abs", "sd", "sd_rel")) %>%
  ggplot() +
  geom_vline(xintercept = 0) +
  geom_hline(yintercept = 20) +
  geom_vline(xintercept = c(10), col = "red") +
  geom_path(aes(pCO2_delta, dep_grid, col = as.factor(tau_factor))) +
  scale_y_reverse() +
  scale_color_discrete(name = "Tau factor") +
  labs(x = expression(Delta ~ italic(p)*CO[2] ~ (µatm)), y = "Depth intervals (1m)") +
  facet_grid(estimate ~ .)
Offset between up- and downcast. Panel columns: Constant and T-dependent tau. Panel rows from top to bottom: Mean of absolute offset, mean of relative absolute offset, standard deviation of offset, standard deviation of relative offset.

Offset between up- and downcast. Panel columns: Constant and T-dependent tau. Panel rows from top to bottom: Mean of absolute offset, mean of relative absolute offset, standard deviation of offset, standard deviation of relative offset.

Mean offset values calculated for the upper 5 and 20 m of the water column:

tm_grid_shallow_sum %>%
  filter(correction == "pCO2",
         estimate %in% c("mean_abs"),
         tau_factor == parameters$tau_factor_used,
         dep_grid < 5) %>% 
  summarise(dep_lim = 5,
            mean(pCO2_delta))
# A tibble: 1 x 2
  dep_lim `mean(pCO2_delta)`
    <dbl>              <dbl>
1       5               2.38
tm_grid_shallow_sum %>%
  filter(correction == "pCO2",
         estimate %in% c("mean_abs"),
         tau_factor == parameters$tau_factor_used,
         dep_grid < 20) %>% 
  summarise(dep_lim = 20,
            mean(pCO2_delta))
# A tibble: 1 x 2
  dep_lim `mean(pCO2_delta)`
    <dbl>              <dbl>
1      20               7.16

Likewise, we analyse the offset from the pCO2 reference value:

tm_equi_delta_sum %>%
  filter(estimate %in% c("mean_abs", "mean_rel_abs")) %>%
  ggplot(aes(tau_factor, dpCO2, linetype = correction, shape = correction)) +
  geom_point() +
  geom_hline(yintercept = 0) +
  labs(x = "Tau factor", y = expression(Mean ~ Delta ~ italic(p)*CO[2])) +
  facet_wrap( ~ estimate)
Mean pCO~2~ offset from reference values as a function of the factor applied to tau. The lines between discrete tau factors result from the same analysis performed with high resolution of the tau factor. Left Panel: Mean absolute offset (µatm). Right panel: Mean relative offset (% of absolute value).

Mean pCO2 offset from reference values as a function of the factor applied to tau. The lines between discrete tau factors result from the same analysis performed with high resolution of the tau factor. Left Panel: Mean absolute offset (µatm). Right panel: Mean relative offset (% of absolute value).

5.5 Summary plots

i_tau_factor <- "1"

cast_dep <- tm %>%
  filter(tau_factor == i_tau_factor) %>%
  pivot_longer(c(dep, pCO2_corr, pCO2),
               names_to = "parameter",
               values_to = "value")

cast_dep_equi <- tm_pCO2_equi %>%
  pivot_longer(c(dep, pCO2_corr), names_to = "parameter", values_to = "value")

tm_sub <- tm %>%
  filter(tau_factor == i_tau_factor)

tm_grid_sub <- tm_grid %>%
  filter(tau_factor == i_tau_factor)

max_duration <- round(max(cast_dep$duration) / 1000, 0) * 1000

pdf(
  file = here::here("output/Plots/response_time",
                    "all_plots.pdf"),
  onefile = TRUE,
  width = 7,
  height = 10
)

for (i_ID in unique(tm$ID)) {
  for (i_station in unique(tm$station)) {
    # i_ID <- unique(cast_dep$ID)[1]
    # i_station <- unique(cast_dep$station)[1]
    i_ID
    i_station
    
    if (nrow(cast_dep %>% filter(ID == i_ID, station == i_station)) > 0) {
      if (nrow(tm %>% filter(ID == i_ID, station == i_station)) > 0) {
        if (nrow(tm_grid %>% filter(ID == i_ID, station == i_station)) > 0) {
          cast_dep_equi_sub <- cast_dep_equi %>%
            filter(ID == i_ID,
                   station == i_station)
          
          p_time_series <- cast_dep %>%
            filter(ID == i_ID,
                   station == i_station) %>%
            ggplot(aes(duration, value, col = phase)) +
            geom_point(size = 0.5) +
            geom_point(data = cast_dep_equi_sub,
                       aes(duration, value),
                       col = "black") +
            scale_y_reverse() +
            scale_x_continuous(breaks = seq(0, 6000, 500)) +
            labs(title = str_c("ID: ", i_ID, " | Station: ", i_station)) +
            facet_grid(parameter ~ ., scales = "free_y") +
            theme_bw()
          
          p_profile <- tm_sub %>%
            filter(ID == i_ID,
                   station == i_station,
                   phase %in% c("up", "down")) %>%
            ggplot() +
            geom_path(aes(pCO2_corr, dep, linetype = phase, col = "raw")) +
            geom_path(aes(pCO2, dep, linetype = phase, col = "corrected")) +
            scale_y_reverse() +
            coord_cartesian(ylim = c(25, 0), xlim = c(70, 250)) +
            scale_color_brewer(palette = "Set1",
                               name = "",
                               guide = FALSE) +
            scale_linetype(guide = FALSE) +
            labs(y = "Depth [m]",
                 x = "pCO2",
                 title = "full res")
          
          
          tm_pCO2_equi_grid_sub <- tm_pCO2_equi_grid %>%
            filter(ID == i_ID,
                   station == i_station)
          
          p_profile_grid <- tm_grid_sub %>%
            filter(ID == i_ID,
                   station == i_station) %>%
            arrange(dep_grid) %>%
            ggplot() +
            geom_path(aes(down, dep_grid, col = correction, linetype = "down")) +
            geom_path(aes(up, dep_grid, col = correction, linetype = "up")) +
            geom_point(data = tm_pCO2_equi_grid_sub, aes(pCO2_equi, dep_grid)) +
            scale_y_reverse() +
            coord_cartesian(ylim = c(25, 0), xlim = c(70, 250)) +
            scale_linetype(name = "cast") +
            scale_color_brewer(palette = "Set1", direction = -1) +
            labs(x = "pCO2",
                 title = "1m grid") +
            theme(axis.title.y = element_blank(),
                  axis.text.y = element_blank())
          
          
          p_delta_abs <- tm_grid %>%
            filter(ID == i_ID,
                   station == i_station,
                   correction == "pCO2") %>%
            arrange(dep_grid) %>%
            ggplot(aes(pCO2_delta, dep_grid, col = as.factor(tau_factor))) +
            geom_path() +
            geom_point() +
            scale_y_reverse(breaks = seq(0, 40, 2)) +
            scale_color_discrete(name = "tau factor", guide = FALSE) +
            labs(x = "delta pCO2 [µatm]", y = "Depth [m]") +
            geom_vline(xintercept = 0) +
            geom_vline(xintercept = c(-10, 10), col = "red")
          
          p_delta_rel <- tm_grid %>%
            filter(ID == i_ID,
                   station == i_station,
                   correction == "pCO2") %>%
            arrange(dep_grid) %>%
            ggplot(aes(pCO2_delta_rel, dep_grid, col = as.factor(tau_factor))) +
            geom_path() +
            geom_point() +
            scale_y_reverse(breaks = seq(0, 40, 2)) +
            scale_color_discrete(name = "tau factor") +
            labs(x = "delta pCO2 [% of absolute value]", y = "Depth [m]") +
            geom_vline(xintercept = 0) +
            geom_vline(xintercept = c(-10, 10), col = "red") +
            theme(axis.title.y = element_blank(),
                  axis.text.y = element_blank())
          
          print(p_time_series / (p_profile |
                                   p_profile_grid) / (p_delta_abs | p_delta_rel))
          rm(p_time_series,
             p_profile,
             p_profile_grid,
             p_delta_abs,
             p_delta_rel)
          
        }
      }
    }
    
    
    
  }
}

dev.off()

rm(
  cast_dep,
  cast_dep_equi,
  cast_dep_equi_sub,
  i_station,
  i_ID,
  max_duration,
  tm_pCO2_equi_grid_sub
)

6 Conclusion

  • Taking the temperature dependence of tau into account resulted in a slightly better agreement between up- and downcast, as well as downcast and reference value (Results not included above anymore)
  • For most quality metrics we find improved agreement for slightly positive tau factor ranging from 1.04 - 1.24
  • Still, the improvement by using a tau factor other then 1 is much lower than the standard deviation of the offset. Therefore, we assume that the improvement is not significant and choose to avoid the application of a tau factor for further analysis

7 Correct entire data set

Finally, the response time correction was applied to the full data set (not only profile data) based on the optimum parameterization determined above.

# Response time correction approach after Bittig et al, 2018
RT_corr <- function(c1, c0, dt, tau) {
  (1 / (2 * ((1 + (
    2 * tau / dt
  )) ^ (-1)))) * (c1 - (1 - (2 * ((
    1 + (2 * tau / dt)
  ) ^ (
    -1
  )))) * c0)
}



tm <-
  read_csv(
    here::here(
      "data/intermediate/_merged_data_files/merging_interpolation",
      "tm.csv"
    ),
    col_types = cols(
      ID = col_character(),
      pCO2_analog = col_double(),
      pCO2_corr = col_double(),
      Zero = col_factor(),
      Flush = col_factor(),
      Zero_counter = col_integer(),
      deployment = col_integer(),
      duration = col_double(),
      mixing = col_character(),
      lat = col_double(),
      lon = col_double()
    )
  )

tm <- tm %>%
  group_by(ID, station) %>%
  mutate(
    duration = as.numeric(date_time - min(date_time)),
    pump_power = if_else(date_time < ymd_hms("2018-07-17;13:08:34"), "1W", "8W")
  ) %>%
  arrange(date_time)


tau_fit <-
  read_csv(here::here(
    "data/intermediate/_merged_data_files/response_time",
    "tau_fit.csv"
  ))



# Assign T-dependent response time (tau) values
tau_fit <- tau_fit %>%
  rename(tau_intercept = `(Intercept)`, tau_slope = tem)

tm <- full_join(tm, tau_fit)

tm <- tm %>%
  mutate(tau = tau_intercept + tau_slope * tem) %>%
  select(-tau_intercept, -tau_slope)


# Prepare data set for RT correction

tm <- tm %>%
  group_by(ID, station) %>%
  arrange(date_time) %>%
  mutate(dt = as.numeric(as.character(date_time - lag(date_time)))) %>%
  ungroup()

# determine measurement frequency

freq <- tm %>%
  filter(dt < 13) %>%
  group_by(ID) %>%
  summarise(dt_mean = round(mean(dt, na.rm = TRUE), 0))

tm <- full_join(tm, freq)

# apply tau factor used
tm <- expand_grid(tm, tau_factor = parameters$tau_factor_used)

tm <- tm %>%
  mutate(tau_test = tau * tau_factor)


# Apply RT correction to entire data set

for (i_ID in unique(tm$ID)) {
  #i_ID <- "180716"
  
  freq_sub <- freq %>% filter(ID == i_ID) %>% pull(dt_mean)
  
  window <- 30 / freq_sub
  rolling_mean   <-
    rollify(~ mean(.x, na.rm = TRUE), window = window)
  
  tm_sub <- tm %>%
    filter(ID == i_ID) %>%
    group_by(station) %>%
    mutate(
      pCO2_RT = RT_corr(pCO2_corr, lag(pCO2_corr), dt, tau_test),
      pCO2_RT = if_else(pCO2_RT %in% c(Inf, -Inf), NaN, pCO2_RT),
      window = window,
      pCO2 = rolling_mean(pCO2_RT)
    ) %>%
    ungroup()
  
  # time shift RT corrected data
  shift <- as.integer(as.character(window / 2))
  
  tm_sub <- tm_sub %>%
    group_by(station) %>%
    mutate(pCO2 = lead(pCO2, shift)) %>%
    ungroup()
  
  
  if (exists("tm_corr")) {
    tm_corr <- bind_rows(tm_corr, tm_sub)
  }
  else{
    tm_corr <- tm_sub
  }
  
  rm(tm_sub, freq_sub, rolling_mean, shift, window)
  
}


rm(RT_corr, i_ID, freq)

tm_corr <- tm_corr %>%
  select(-c(tau, tau_factor, tau_test, window))

rm(tm)

7.1 Histogram

Below, histograms of RT corrected pCO2 are shown, for selected ranges of pCO2.

tm_corr %>% 
  ggplot(aes(pCO2)) +
  geom_histogram()

tm_corr %>% 
  filter(pCO2 < 0) %>% 
  ggplot(aes(pCO2)) +
  geom_histogram()

tm_corr %>% 
  filter(pCO2 < 200,
         pCO2 > 0) %>% 
  ggplot(aes(pCO2)) +
  geom_histogram()

8 Write summary file

Response time corrected pCO2 data with absolute values >50 are written to file for further analysis. Negative pCO2 readings are considered artefacts of the response time correction.

tm_corr %>%
  filter(pCO2 > 50) %>% 
  write_csv(
    here::here(
      "data/intermediate/_merged_data_files/response_time",
      "tm_RT_all.csv"
    )
  )

9 Sensitivity considerations

A change in DIC of 1 µmol kg-1 corresponds to a change in pCO2 of around 1 µatm, in the Central Baltic Sea at a pCO2 of around 100 µatm (summertime conditions).

df <- data.frame(cbind((c(1720)),
                       (c(7))))

Tem <- seq(5, 25, 5)
pCO2 <- seq(50, 500, 20)

df <- merge(df, Tem)
names(df) <- c("AT", "S", "Tem")

df <- merge(df, pCO2)
names(df) <- c("AT", "S", "Tem", "pCO2")

df$AT <- df$AT * 1e-6

df$DIC <-
  carb(
    flag = 24,
    var1 = df$pCO2,
    var2 = df$AT,
    S = df$S,
    T = df$Tem,
    k1k2 = "m10",
    kf = "dg",
    pHscale = "T"
  )[, 16]
df$pCO2.corr <-
  carb(
    flag = 15,
    var1 = df$AT,
    var2 = df$DIC,
    S = df$S,
    T = df$Tem,
    k1k2 = "m10",
    kf = "dg",
    pHscale = "T"
  )[, 9]

df$pCO2.2 <- df$pCO2.corr + 25
df$DIC.2 <-
  carb(
    flag = 24,
    var1 = df$pCO2.2,
    var2 = df$AT,
    S = df$S,
    T = df$Tem,
    k1k2 = "m10",
    kf = "dg",
    pHscale = "T"
  )[, 16]


df$ratio <- (df$pCO2.2 - df$pCO2.corr) / (df$DIC.2 * 1e6 - df$DIC * 1e6)

df %>%
  ggplot(aes(pCO2, ratio, col = as.factor(Tem))) +
  geom_line() +
  scale_color_viridis_d(option = "C", name = "Tem [°C]") +
  labs(x = expression(italic(p)*CO[2] ~ (µatm)),
       y = expression(Delta ~ italic(p)*CO[2] ~ "/" ~ Delta ~ DIC ~ (µatm ~ µmol ^ {
         -1
       } ~ kg))) +
  scale_y_continuous(limits = c(0, 8), breaks = seq(0, 10, 1))
pCO~2~ sensitivity to changes in DIC.

pCO2 sensitivity to changes in DIC.

rm(df, Tem, pCO2)

sessionInfo()
R version 4.0.3 (2020-10-10)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 18363)

Matrix products: default

locale:
[1] LC_COLLATE=English_Germany.1252  LC_CTYPE=English_Germany.1252   
[3] LC_MONETARY=English_Germany.1252 LC_NUMERIC=C                    
[5] LC_TIME=English_Germany.1252    

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] patchwork_1.1.1   tibbletime_0.1.6  lubridate_1.7.9.2 broom_0.7.3      
 [5] seacarb_3.2.14    oce_1.2-0         gsw_1.0-5         testthat_3.0.1   
 [9] forcats_0.5.0     stringr_1.4.0     dplyr_1.0.2       purrr_0.3.4      
[13] readr_1.4.0       tidyr_1.1.2       tibble_3.0.4      ggplot2_3.3.3    
[17] tidyverse_1.3.0   workflowr_1.6.2  

loaded via a namespace (and not attached):
 [1] httr_1.4.2         viridisLite_0.3.0  jsonlite_1.7.2     splines_4.0.3     
 [5] here_1.0.1         modelr_0.1.8       assertthat_0.2.1   highr_0.8         
 [9] cellranger_1.1.0   yaml_2.2.1         pillar_1.4.7       backports_1.2.1   
[13] lattice_0.20-41    glue_1.4.2         digest_0.6.27      RColorBrewer_1.1-2
[17] promises_1.1.1     rvest_0.3.6        colorspace_2.0-0   htmltools_0.5.0   
[21] httpuv_1.5.4       Matrix_1.2-18      pkgconfig_2.0.3    haven_2.3.1       
[25] scales_1.1.1       whisker_0.4        later_1.1.0.1      git2r_0.27.1      
[29] mgcv_1.8-33        generics_0.1.0     farver_2.0.3       ellipsis_0.3.1    
[33] withr_2.3.0        cli_2.2.0          magrittr_2.0.1     crayon_1.3.4      
[37] readxl_1.3.1       evaluate_0.14      ps_1.5.0           fs_1.5.0          
[41] fansi_0.4.1        nlme_3.1-149       xml2_1.3.2         tools_4.0.3       
[45] hms_0.5.3          lifecycle_0.2.0    munsell_0.5.0      reprex_0.3.0      
[49] compiler_4.0.3     rlang_0.4.10       grid_4.0.3         rstudioapi_0.13   
[53] labeling_0.4.2     rmarkdown_2.6      gtable_0.3.0       DBI_1.1.0         
[57] R6_2.5.0           knitr_1.30         utf8_1.1.4         rprojroot_2.0.2   
[61] stringi_1.5.3      Rcpp_1.0.5         vctrs_0.3.6        dbplyr_2.0.0      
[65] tidyselect_1.1.0   xfun_0.19