An introduction to climate forecasting with deep studying

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With all that is occurring on this planet as of late, is it frivolous to speak about climate prediction? Requested within the twenty first century, that is sure to be a rhetorical query. Within the Nineteen Thirties, when German poet Bertolt Brecht wrote the well-known strains:

Was sind das für Zeiten, wo
Ein Gespräch über Bäume quick ein Verbrechen ist
Weil es ein Schweigen über so viele Untaten einschließt!

(“What sort of occasions are these, the place a dialog about timber is sort of a criminal offense, for it means silence about so many atrocities!”),

he couldn’t have anticipated the responses he would get within the second half of that century, with timber symbolizing, in addition to actually falling sufferer to, environmental air pollution and local weather change.

At this time, no prolonged justification is required as to why prediction of atmospheric states is important: As a result of world warming, frequency and depth of extreme climate situations – droughts, wildfires, hurricanes, heatwaves – have risen and can proceed to rise. And whereas correct forecasts don’t change these occasions per se, they represent important data in mitigating their penalties. This goes for atmospheric forecasts on all scales: from so-called “nowcasting” (working on a spread of about six hours), over medium-range (three to 5 days) and sub-seasonal (weekly/month-to-month), to local weather forecasts (involved with years and many years). Medium-range forecasts particularly are extraordinarily essential in acute catastrophe prevention.

This submit will present how deep studying (DL) strategies can be utilized to generate atmospheric forecasts, utilizing a newly printed benchmark dataset(Rasp et al. 2020). Future posts could refine the mannequin used right here and/or focus on the position of DL (“AI”) in mitigating local weather change – and its implications – extra globally.

That mentioned, let’s put the present endeavor in context. In a manner, now we have right here the standard dejà vu of utilizing DL as a black-box-like, magic instrument on a job the place human data was once required. After all, this characterization is overly dichotomizing; many selections are made in creating DL fashions, and efficiency is essentially constrained by out there algorithms – which can, or could not, match the area to be modeled to a ample diploma.

In case you’ve began studying about picture recognition quite lately, chances are you’ll nicely have been utilizing DL strategies from the outset, and never have heard a lot concerning the wealthy set of characteristic engineering strategies developed in pre-DL picture recognition. Within the context of atmospheric prediction, then, let’s start by asking: How on this planet did they try this earlier than?

Numerical climate prediction in a nutshell

It’s not like machine studying and/or statistics will not be already utilized in numerical climate prediction – quite the opposite. For instance, each mannequin has to start out from someplace; however uncooked observations will not be suited to direct use as preliminary situations. As an alternative, they need to be assimilated to the four-dimensional grid over which mannequin computations are carried out. On the different finish, specifically, mannequin output, statistical post-processing is used to refine the predictions. And really importantly, ensemble forecasts are employed to find out uncertainty.

That mentioned, the mannequin core, the half that extrapolates into the long run atmospheric situations noticed as we speak, relies on a set of differential equations, the so-called primitive equations, which might be because of the conservation legal guidelines of momentum, power, and mass. These differential equations can’t be solved analytically; quite, they need to be solved numerically, and that on a grid of decision as excessive as potential. In that mild, even deep studying may seem as simply “reasonably resource-intensive” (dependent, although, on the mannequin in query). So how, then, may a DL method look?

Deep studying fashions for climate prediction

Accompanying the benchmark dataset they created, Rasp et al.(Rasp et al. 2020) present a set of notebooks, together with one demonstrating using a easy convolutional neural community to foretell two of the out there atmospheric variables, 500hPa geopotential and 850hPa temperature. Right here 850hPa temperature is the (spatially various) temperature at a repair atmospheric peak of 850hPa (~ 1.5 kms) ; 500hPa geopotential is proportional to the (once more, spatially various) altitude related to the strain degree in query (500hPa).

For this job, two-dimensional convnets, as often employed in picture processing, are a pure match: Picture width and peak map to longitude and latitude of the spatial grid, respectively; goal variables seem as channels. On this structure, the time collection character of the information is basically misplaced: Each pattern stands alone, with out dependency on both previous or current. On this respect, in addition to given its dimension and ease, the convnet introduced under is barely a toy mannequin, meant to introduce the method in addition to the applying general. It might additionally function a deep studying baseline, together with two different kinds of baseline generally utilized in numerical climate prediction launched under.

Instructions on how you can enhance on that baseline are given by latest publications. Weyn et al.(Weyn, Durran, and Caruana, n.d.), along with making use of extra geometrically-adequate spatial preprocessing, use a U-Internet-based structure as an alternative of a plain convnet. Rasp and Thuerey (Rasp and Thuerey 2020), constructing on a totally convolutional, high-capacity ResNet structure, add a key new procedural ingredient: pre-training on local weather fashions. With their methodology, they can not simply compete with bodily fashions, but in addition, present proof of the community studying about bodily construction and dependencies. Sadly, compute services of this order will not be out there to the common particular person, which is why we’ll content material ourselves with demonstrating a easy toy mannequin. Nonetheless, having seen a easy mannequin in motion, in addition to the kind of knowledge it really works on, ought to assist loads in understanding how DL can be utilized for climate prediction.

Dataset

Weatherbench was explicitly created as a benchmark dataset and thus, as is widespread for this species, hides lots of preprocessing and standardization effort from the consumer. Atmospheric knowledge can be found on an hourly foundation, starting from 1979 to 2018, at totally different spatial resolutions. Relying on decision, there are about 15 to twenty measured variables, together with temperature, geopotential, wind pace, and humidity. Of those variables, some can be found at a number of strain ranges. Thus, our instance makes use of a small subset of obtainable “channels.” To save lots of storage, community and computational sources, it additionally operates on the smallest out there decision.

This submit is accompanied by executable code on Google Colaboratory, which shouldn’t simply render pointless any copy-pasting of code snippets but in addition, enable for uncomplicated modification and experimentation.

To learn in and extract the information, saved as NetCDF information, we use tidync, a high-level package deal constructed on prime of ncdf4 and RNetCDF. In any other case, availability of the standard “TensorFlow household” in addition to a subset of tidyverse packages is assumed.

As already alluded to, our instance makes use of two spatio-temporal collection: 500hPa geopotential and 850hPa temperature. The next instructions will obtain and unpack the respective units of by-year information, for a spatial decision of 5.625 levels:

obtain.file("https://dataserv.ub.tum.de/s/m1524895/obtain?path=%2F5.625degpercent2Ftemperature_850&information=temperature_850_5.625deg.zip",
              "temperature_850_5.625deg.zip")
unzip("temperature_850_5.625deg.zip", exdir = "temperature_850")

obtain.file("https://dataserv.ub.tum.de/s/m1524895/obtain?path=%2F5.625degpercent2Fgeopotential_500&information=geopotential_500_5.625deg.zip",
              "geopotential_500_5.625deg.zip")
unzip("geopotential_500_5.625deg.zip", exdir = "geopotential_500")

Inspecting a type of information’ contents, we see that its knowledge array is structured alongside three dimensions, longitude (64 totally different values), latitude (32) and time (8760). The information itself is z, the geopotential.

tidync("geopotential_500/geopotential_500hPa_2015_5.625deg.nc") %>% hyper_array()
Class: tidync_data (checklist of tidync knowledge arrays)
Variables (1): 'z'
Dimension (3): lon,lat,time (64, 32, 8760)
Supply: /[...]/geopotential_500/geopotential_500hPa_2015_5.625deg.nc

Extraction of the information array is as simple as telling tidync to learn the primary within the checklist of arrays:

z500_2015 <- (tidync("geopotential_500/geopotential_500hPa_2015_5.625deg.nc") %>%
                hyper_array())[[1]]

dim(z500_2015)
[1] 64 32 8760

Whereas we delegate additional introduction to tidync to a complete weblog submit on the ROpenSci web site, let’s at the very least have a look at a fast visualization, for which we choose the very first time level. (Extraction and visualization code is analogous for 850hPa temperature.)

picture(z500_2015[ , , 1],
      col = hcl.colours(20, "viridis"), # for temperature, the colour scheme used is YlOrRd 
      xaxt = 'n',
      yaxt = 'n',
      most important = "500hPa geopotential"
)

The maps present how strain and temperature strongly depend upon latitude. Moreover, it’s simple to identify the atmospheric waves:


Spatial distribution of 500hPa geopotential and 850 hPa temperature for 2015/01/01 0:00h.

Determine 1: Spatial distribution of 500hPa geopotential and 850 hPa temperature for 2015/01/01 0:00h.

For coaching, validation and testing, we select consecutive years: 2015, 2016, and 2017, respectively.

z500_train <- (tidync("geopotential_500/geopotential_500hPa_2015_5.625deg.nc") %>% hyper_array())[[1]]

t850_train <- (tidync("temperature_850/temperature_850hPa_2015_5.625deg.nc") %>% hyper_array())[[1]]

z500_valid <- (tidync("geopotential_500/geopotential_500hPa_2016_5.625deg.nc") %>% hyper_array())[[1]]

t850_valid <- (tidync("temperature_850/temperature_850hPa_2016_5.625deg.nc") %>% hyper_array())[[1]]

z500_test <- (tidync("geopotential_500/geopotential_500hPa_2017_5.625deg.nc") %>% hyper_array())[[1]]

t850_test <- (tidync("temperature_850/temperature_850hPa_2017_5.625deg.nc") %>% hyper_array())[[1]]

Since geopotential and temperature might be handled as channels, we concatenate the corresponding arrays. To remodel the information into the format wanted for photographs, a permutation is important:

train_all <- abind::abind(z500_train, t850_train, alongside = 4)
train_all <- aperm(train_all, perm = c(3, 2, 1, 4))
dim(train_all)
[1] 8760 32 64 2

All knowledge might be standardized based on imply and commonplace deviation as obtained from the coaching set:

level_means <- apply(train_all, 4, imply)
level_sds <- apply(train_all, 4, sd)

spherical(level_means, 2)
54124.91  274.8

In phrases, the imply geopotential peak (see footnote 5 for extra on this time period), as measured at an isobaric floor of 500hPa, quantities to about 5400 metres, whereas the imply temperature on the 850hPa degree approximates 275 Kelvin (about 2 levels Celsius).

practice <- train_all
practice[, , , 1] <- (practice[, , , 1] - level_means[1]) / level_sds[1]
practice[, , , 2] <- (practice[, , , 2] - level_means[2]) / level_sds[2]

valid_all <- abind::abind(z500_valid, t850_valid, alongside = 4)
valid_all <- aperm(valid_all, perm = c(3, 2, 1, 4))

legitimate <- valid_all
legitimate[, , , 1] <- (legitimate[, , , 1] - level_means[1]) / level_sds[1]
legitimate[, , , 2] <- (legitimate[, , , 2] - level_means[2]) / level_sds[2]

test_all <- abind::abind(z500_test, t850_test, alongside = 4)
test_all <- aperm(test_all, perm = c(3, 2, 1, 4))

take a look at <- test_all
take a look at[, , , 1] <- (take a look at[, , , 1] - level_means[1]) / level_sds[1]
take a look at[, , , 2] <- (take a look at[, , , 2] - level_means[2]) / level_sds[2]

We’ll try and predict three days forward.

Now all that is still to be finished is assemble the precise datasets.

batch_size <- 32

train_x <- practice %>%
  tensor_slices_dataset() %>%
  dataset_take(dim(practice)[1] - lead_time)

train_y <- practice %>%
  tensor_slices_dataset() %>%
  dataset_skip(lead_time)

train_ds <- zip_datasets(train_x, train_y) %>%
  dataset_shuffle(buffer_size = dim(practice)[1] - lead_time) %>%
  dataset_batch(batch_size = batch_size, drop_remainder = TRUE)

valid_x <- legitimate %>%
  tensor_slices_dataset() %>%
  dataset_take(dim(legitimate)[1] - lead_time)

valid_y <- legitimate %>%
  tensor_slices_dataset() %>%
  dataset_skip(lead_time)

valid_ds <- zip_datasets(valid_x, valid_y) %>%
  dataset_batch(batch_size = batch_size, drop_remainder = TRUE)

test_x <- take a look at %>%
  tensor_slices_dataset() %>%
  dataset_take(dim(take a look at)[1] - lead_time)

test_y <- take a look at %>%
  tensor_slices_dataset() %>%
  dataset_skip(lead_time)

test_ds <- zip_datasets(test_x, test_y) %>%
  dataset_batch(batch_size = batch_size, drop_remainder = TRUE)

Let’s proceed to defining the mannequin.

Primary CNN with periodic convolutions

The mannequin is an easy convnet, with one exception: As an alternative of plain convolutions, it makes use of barely extra refined ones that “wrap round” longitudinally.

periodic_padding_2d <- perform(pad_width,
                                identify = NULL) {
  
  keras_model_custom(identify = identify, perform(self) {
    self$pad_width <- pad_width
    
    perform (x, masks = NULL) {
      x <- if (self$pad_width == 0) {
        x
      } else {
        lon_dim <- dim(x)[3]
        pad_width <- tf$forged(self$pad_width, tf$int32)
        # wrap round for longitude
        tf$concat(checklist(x[, ,-pad_width:lon_dim,],
                       x,
                       x[, , 1:pad_width,]),
                  axis = 2L) %>%
          tf$pad(checklist(
            checklist(0L, 0L),
            # zero-pad for latitude
            checklist(pad_width, pad_width),
            checklist(0L, 0L),
            checklist(0L, 0L)
          ))
      }
    }
  })
}

periodic_conv_2d <- perform(filters,
                             kernel_size,
                             identify = NULL) {
  
  keras_model_custom(identify = identify, perform(self) {
    self$padding <- periodic_padding_2d(pad_width = (kernel_size - 1) / 2)
    self$conv <-
      layer_conv_2d(filters = filters,
                    kernel_size = kernel_size,
                    padding = 'legitimate')
    
    perform (x, masks = NULL) {
      x %>% self$padding() %>% self$conv()
    }
  })
}

For our functions of creating a deep-learning baseline that’s quick to coach, CNN structure and parameter defaults are chosen to be easy and average, respectively:

periodic_cnn <- perform(filters = c(64, 64, 64, 64, 2),
                         kernel_size = c(5, 5, 5, 5, 5),
                         dropout = rep(0.2, 5),
                         identify = NULL) {
  
  keras_model_custom(identify = identify, perform(self) {
    
    self$conv1 <-
      periodic_conv_2d(filters = filters[1], kernel_size = kernel_size[1])
    self$act1 <- layer_activation_leaky_relu()
    self$drop1 <- layer_dropout(price = dropout[1])
    self$conv2 <-
      periodic_conv_2d(filters = filters[2], kernel_size = kernel_size[2])
    self$act2 <- layer_activation_leaky_relu()
    self$drop2 <- layer_dropout(price =dropout[2])
    self$conv3 <-
      periodic_conv_2d(filters = filters[3], kernel_size = kernel_size[3])
    self$act3 <- layer_activation_leaky_relu()
    self$drop3 <- layer_dropout(price = dropout[3])
    self$conv4 <-
      periodic_conv_2d(filters = filters[4], kernel_size = kernel_size[4])
    self$act4 <- layer_activation_leaky_relu()
    self$drop4 <- layer_dropout(price = dropout[4])
    self$conv5 <-
      periodic_conv_2d(filters = filters[5], kernel_size = kernel_size[5])
    
    perform (x, masks = NULL) {
      x %>%
        self$conv1() %>%
        self$act1() %>%
        self$drop1() %>%
        self$conv2() %>%
        self$act2() %>%
        self$drop2() %>%
        self$conv3() %>%
        self$act3() %>%
        self$drop3() %>%
        self$conv4() %>%
        self$act4() %>%
        self$drop4() %>%
        self$conv5()
    }
  })
}

mannequin <- periodic_cnn()

Coaching

In that very same spirit of “default-ness,” we practice with MSE loss and Adam optimizer.

loss <- tf$keras$losses$MeanSquaredError(discount = tf$keras$losses$Discount$SUM)
optimizer <- optimizer_adam()

train_loss <- tf$keras$metrics$Imply(identify='train_loss')

valid_loss <- tf$keras$metrics$Imply(identify='test_loss')

train_step <- perform(train_batch) {

  with (tf$GradientTape() %as% tape, {
    predictions <- mannequin(train_batch[[1]])
    l <- loss(train_batch[[2]], predictions)
  })

  gradients <- tape$gradient(l, mannequin$trainable_variables)
  optimizer$apply_gradients(purrr::transpose(checklist(
    gradients, mannequin$trainable_variables
  )))

  train_loss(l)

}

valid_step <- perform(valid_batch) {
  predictions <- mannequin(valid_batch[[1]])
  l <- loss(valid_batch[[2]], predictions)
  
  valid_loss(l)
}

training_loop <- tf_function(autograph(perform(train_ds, valid_ds, epoch) {
  
  for (train_batch in train_ds) {
    train_step(train_batch)
  }
  
  for (valid_batch in valid_ds) {
    valid_step(valid_batch)
  }
  
  tf$print("MSE: practice: ", train_loss$consequence(), ", validation: ", valid_loss$consequence()) 
    
}))

Depicted graphically, we see that the mannequin trains nicely, however extrapolation doesn’t surpass a sure threshold (which is reached early, after coaching for simply two epochs).


MSE per epoch on training and validation sets.

Determine 2: MSE per epoch on coaching and validation units.

This isn’t too shocking although, given the mannequin’s architectural simplicity and modest dimension.

Analysis

Right here, we first current two different baselines, which – given a extremely complicated and chaotic system just like the ambiance – could sound irritatingly easy and but, be fairly exhausting to beat. The metric used for comparability is latitudinally weighted root-mean-square error. Latitudinal weighting up-weights the decrease latitudes and down-weights the higher ones.

deg2rad <- perform(d) {
  (d / 180) * pi
}

lats <- tidync("geopotential_500/geopotential_500hPa_2015_5.625deg.nc")$transforms$lat %>%
  choose(lat) %>%
  pull()

lat_weights <- cos(deg2rad(lats))
lat_weights <- lat_weights / imply(lat_weights)

weighted_rmse <- perform(forecast, ground_truth) {
  error <- (forecast - ground_truth) ^ 2
  for (i in seq_along(lat_weights)) {
    error[, i, ,] <- error[, i, ,] * lat_weights[i]
  }
  apply(error, 4, imply) %>% sqrt()
}

Baseline 1: Weekly climatology

Generally, climatology refers to long-term averages computed over outlined time ranges. Right here, we first calculate weekly averages based mostly on the coaching set. These averages are then used to forecast the variables in query for the time interval used as take a look at set.

The 1st step makes use of tidync, ncmeta, RNetCDF and lubridate to compute weekly averages for 2015, following the ISO week date system.

train_file <- "geopotential_500/geopotential_500hPa_2015_5.625deg.nc"

times_train <- (tidync(train_file) %>% activate("D2") %>% hyper_array())$time

time_unit_train <- ncmeta::nc_atts(train_file, "time") %>%
  tidyr::unnest(cols = c(worth)) %>%
  dplyr::filter(identify == "models")

time_parts_train <- RNetCDF::utcal.nc(time_unit_train$worth, times_train)

iso_train <- ISOdate(
  time_parts_train[, "year"],
  time_parts_train[, "month"],
  time_parts_train[, "day"],
  time_parts_train[, "hour"],
  time_parts_train[, "minute"],
  time_parts_train[, "second"]
)

isoweeks_train <- map(iso_train, isoweek) %>% unlist()

train_by_week <- apply(train_all, c(2, 3, 4), perform(x) {
  tapply(x, isoweeks_train, perform(y) {
    imply(y)
  })
})

dim(train_by_week)
53 32 64 2

Step two then runs via the take a look at set, mapping dates to corresponding ISO weeks and associating the weekly averages from the coaching set:

test_file <- "geopotential_500/geopotential_500hPa_2017_5.625deg.nc"

times_test <- (tidync(test_file) %>% activate("D2") %>% hyper_array())$time

time_unit_test <- ncmeta::nc_atts(test_file, "time") %>%
  tidyr::unnest(cols = c(worth)) %>%
  dplyr::filter(identify == "models")

time_parts_test <- RNetCDF::utcal.nc(time_unit_test$worth, times_test)

iso_test <- ISOdate(
  time_parts_test[, "year"],
  time_parts_test[, "month"],
  time_parts_test[, "day"],
  time_parts_test[, "hour"],
  time_parts_test[, "minute"],
  time_parts_test[, "second"]
)

isoweeks_test <- map(iso_test, isoweek) %>% unlist()

climatology_forecast <- test_all

for (i in 1:dim(climatology_forecast)[1]) {
  week <- isoweeks_test[i]
  lookup <- train_by_week[week, , , ]
  climatology_forecast[i, , ,] <- lookup
}

For this baseline, the latitudinally-weighted RMSE quantities to roughly 975 for geopotential and 4 for temperature.

wrmse <- weighted_rmse(climatology_forecast, test_all)
spherical(wrmse, 2)
974.50   4.09

Baseline 2: Persistence forecast

The second baseline generally used makes a simple assumption: Tomorrow’s climate is as we speak’s climate, or, in our case: In three days, issues might be similar to they’re now.

Computation for this metric is sort of a one-liner. And because it seems, for the given lead time (three days), efficiency isn’t too dissimilar from obtained by the use of weekly climatology:

persistence_forecast <- test_all[1:(dim(test_all)[1] - lead_time), , ,]

test_period <- test_all[(lead_time + 1):dim(test_all)[1], , ,]

wrmse <- weighted_rmse(persistence_forecast, test_period)

spherical(wrmse, 2)
937.55  4.31

Baseline 3: Easy convnet

How does the easy deep studying mannequin stack up towards these two?

To reply that query, we first must acquire predictions on the take a look at set.

test_wrmses <- knowledge.body()

test_loss <- tf$keras$metrics$Imply(identify = 'test_loss')

test_step <- perform(test_batch, batch_index) {
  predictions <- mannequin(test_batch[[1]])
  l <- loss(test_batch[[2]], predictions)
  
  predictions <- predictions %>% as.array()
  predictions[, , , 1] <- predictions[, , , 1] * level_sds[1] + level_means[1]
  predictions[, , , 2] <- predictions[, , , 2] * level_sds[2] + level_means[2]
  
  wrmse <- weighted_rmse(predictions, test_all[batch_index:(batch_index + 31), , ,])
  test_wrmses <<- test_wrmses %>% bind_rows(c(z = wrmse[1], temp = wrmse[2]))

  test_loss(l)
}

test_iterator <- as_iterator(test_ds)

batch_index <- 0
whereas (TRUE) {
  test_batch <- test_iterator %>% iter_next()
  if (is.null(test_batch))
    break
  batch_index <- batch_index + 1
  test_step(test_batch, as.integer(batch_index))
}

test_loss$consequence() %>% as.numeric()
3821.016

Thus, common loss on the take a look at set parallels that seen on the validation set. As to latitudinally weighted RMSE, it seems to be greater for the DL baseline than for the opposite two:

      z    temp 
1521.47    7.70 

Conclusion

At first look, seeing the DL baseline carry out worse than the others would possibly really feel anticlimactic. But when you consider it, there is no such thing as a should be disillusioned.

For one, given the big complexity of the duty, these heuristics will not be as simple to outsmart. Take persistence: Relying on lead time – how far into the long run we’re forecasting – the wisest guess may very well be that every part will keep the identical. What would you guess the climate will appear like in 5 minutes? — Identical with weekly climatology: Wanting again at how heat it was, at a given location, that very same week two years in the past, doesn’t on the whole sound like a nasty technique.

Second, the DL baseline proven is as primary as it will probably get, architecture- in addition to parameter-wise. Extra refined and highly effective architectures have been developed that not simply by far surpass the baselines, however may even compete with bodily fashions (cf. particularly Rasp and Thuerey (Rasp and Thuerey 2020) already talked about above). Sadly, fashions like that should be skilled on loads of information.

Nonetheless, different weather-related functions (apart from medium-range forecasting, that’s) could also be extra in attain for people within the subject. For these, we hope now we have given a helpful introduction. Thanks for studying!

Rasp, Stephan, Peter D. Dueben, Sebastian Scher, Jonathan A. Weyn, Soukayna Mouatadid, and Nils Thuerey. 2020. WeatherBench: A benchmark dataset for data-driven climate forecasting.” arXiv e-Prints, February, arXiv:2002.00469. https://arxiv.org/abs/2002.00469.
Rasp, Stephan, and Nils Thuerey. 2020. “Purely Knowledge-Pushed Medium-Vary Climate Forecasting Achieves Comparable Ability to Bodily Fashions at Comparable Decision.” https://arxiv.org/abs/2008.08626.
Weyn, Jonathan A., Dale R. Durran, and Wealthy Caruana. n.d. “Bettering Knowledge-Pushed International Climate Prediction Utilizing Deep Convolutional Neural Networks on a Cubed Sphere.” Journal of Advances in Modeling Earth Programs n/a (n/a): e2020MS002109. https://doi.org/10.1029/2020MS002109.

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