An introduction to climate forecasting with deep studying

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With all that is happening on the 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 traces:

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 instances 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 present, no prolonged justification is required as to why prediction of atmospheric states is significant: 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 info 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 vital in acute catastrophe prevention.

This publish will present how deep studying (DL) strategies can be utilized to generate atmospheric forecasts, utilizing a newly revealed benchmark dataset(Rasp et al. 2020). Future posts might refine the mannequin used right here and/or talk about the function 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 means, we now have right here the same old dejà vu of utilizing DL as a black-box-like, magic instrument on a process the place human information was once required. After all, this characterization is overly dichotomizing; many selections are made in creating DL fashions, and efficiency is essentially constrained by accessible algorithms – which can, or might not, match the area to be modeled to a enough diploma.

For those who’ve began studying about picture recognition somewhat not too long ago, it’s possible you’ll effectively have been utilizing DL strategies from the outset, and never have heard a lot in regards to 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 the planet did they try this earlier than?

Numerical climate prediction in a nutshell

It isn’t like machine studying and/or statistics aren’t already utilized in numerical climate prediction – quite the opposite. For instance, each mannequin has to begin from someplace; however uncooked observations aren’t suited to direct use as preliminary situations. As an alternative, they should 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 right now, is predicated on a set of differential equations, the so-called primitive equations, which might be as a result of conservation legal guidelines of momentum, vitality, and mass. These differential equations can’t be solved analytically; somewhat, they should be solved numerically, and that on a grid of decision as excessive as doable. In that gentle, even deep studying might seem as simply “reasonably resource-intensive” (dependent, although, on the mannequin in query). So how, then, might a DL strategy 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 accessible atmospheric variables, 500hPa geopotential and 850hPa temperature. Right here 850hPa temperature is the (spatially various) temperature at a repair atmospheric top of 850hPa (~ 1.5 kms) ; 500hPa geopotential is proportional to the (once more, spatially various) altitude related to the stress stage in query (500hPa).

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

Instructions on how you can enhance on that baseline are given by current 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 completely 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 additionally, present proof of the community studying about bodily construction and dependencies. Sadly, compute amenities of this order aren’t accessible to the typical 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 information it really works on, ought to assist quite a bit 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 numerous preprocessing and standardization effort from the person. Atmospheric information can be found on an hourly foundation, starting from 1979 to 2018, at completely different spatial resolutions. Relying on decision, there are about 15 to twenty measured variables, together with temperature, geopotential, wind velocity, and humidity. Of those variables, some can be found at a number of stress ranges. Thus, our instance makes use of a small subset of obtainable “channels.” To save lots of storage, community and computational assets, it additionally operates on the smallest accessible decision.

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

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

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

obtain.file("https://dataserv.ub.tum.de/s/m1524895/obtain?path=%2F5.625degpercent2Ftemperature_850&recordsdata=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&recordsdata=geopotential_500_5.625deg.zip",
              "geopotential_500_5.625deg.zip")
unzip("geopotential_500_5.625deg.zip", exdir = "geopotential_500")

Inspecting a kind of recordsdata’ contents, we see that its information array is structured alongside three dimensions, longitude (64 completely 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 (record of tidync information arrays)
Variables (1): 'z'
Dimension (3): lon,lat,time (64, 32, 8760)
Supply: /[...]/geopotential_500/geopotential_500hPa_2015_5.625deg.nc

Extraction of the info array is as simple as telling tidync to learn the primary within the record 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 publish on the ROpenSci web site, let’s a minimum of 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',
      principal = "500hPa geopotential"
)

The maps present how stress and temperature strongly rely on 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 can be handled as channels, we concatenate the corresponding arrays. To remodel the info into the format wanted for photos, a permutation is critical:

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 information can be standardized in accordance with imply and customary 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 top (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 stage 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 to predict three days forward.

Now all that is still to be completed 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,
                                title = NULL) {
  
  keras_model_custom(title = title, 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$concat(record(x[, ,-pad_width:lon_dim,],
                       x,
                       x[, , 1:pad_width,]),
                  axis = 2L) %>%
          tf$pad(record(
            record(0L, 0L),
            # zero-pad for latitude
            record(pad_width, pad_width),
            record(0L, 0L),
            record(0L, 0L)
          ))
      }
    }
  })
}

periodic_conv_2d <- perform(filters,
                             kernel_size,
                             title = NULL) {
  
  keras_model_custom(title = title, 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 building 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),
                         title = NULL) {
  
  keras_model_custom(title = title, perform(self) {
    
    self$conv1 <-
      periodic_conv_2d(filters = filters[1], kernel_size = kernel_size[1])
    self$act1 <- layer_activation_leaky_relu()
    self$drop1 <- layer_dropout(charge = dropout[1])
    self$conv2 <-
      periodic_conv_2d(filters = filters[2], kernel_size = kernel_size[2])
    self$act2 <- layer_activation_leaky_relu()
    self$drop2 <- layer_dropout(charge =dropout[2])
    self$conv3 <-
      periodic_conv_2d(filters = filters[3], kernel_size = kernel_size[3])
    self$act3 <- layer_activation_leaky_relu()
    self$drop3 <- layer_dropout(charge = dropout[3])
    self$conv4 <-
      periodic_conv_2d(filters = filters[4], kernel_size = kernel_size[4])
    self$act4 <- layer_activation_leaky_relu()
    self$drop4 <- layer_dropout(charge = 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(title='train_loss')

valid_loss <- tf$keras$metrics$Imply(title='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$apply_gradients(purrr::transpose(record(
    gradients, mannequin$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$print("MSE: practice: ", train_loss$end result(), ", validation: ", valid_loss$end result()) 
    
}))

Depicted graphically, we see that the mannequin trains effectively, 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 measurement.

Analysis

Right here, we first current two different baselines, which – given a extremely advanced and chaotic system just like the environment – might 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

On the whole, climatology refers to long-term averages computed over outlined time ranges. Right here, we first calculate weekly averages primarily based 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(title == "items")

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 by means of 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(title == "items")

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 right now’s climate, or, in our case: In three days, issues can be identical 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 just isn’t too dissimilar from obtained by way 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 in opposition to these two?

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

test_wrmses <- information.body()

test_loss <- tf$keras$metrics$Imply(title = '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$end result() %>% 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 larger 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 may really feel anticlimactic. But when you consider it, there isn’t a must be dissatisfied.

For one, given the large complexity of the duty, these heuristics aren’t 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 the whole lot will keep the identical. What would you guess the climate will appear like in 5 minutes? — Identical with weekly climatology: Trying again at how heat it was, at a given location, that very same week two years in the past, doesn’t generally sound like a foul technique.

Second, the DL baseline proven is as primary as it will possibly 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 must be skilled on quite a bit of knowledge.

Nonetheless, different weather-related purposes (aside from medium-range forecasting, that’s) could also be extra in attain for people within the matter. For these, we hope we now 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 Talent to Bodily Fashions at Comparable Decision.” https://arxiv.org/abs/2008.08626.
Weyn, Jonathan A., Dale R. Durran, and Wealthy Caruana. n.d. “Enhancing 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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