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Vector averaging degrees of wind direction

Beau Larkin 2021-03-05

Description

Wind direction is usually reported in degrees, which presents a problem for calculating an average. For example, if the wind direction for a given period varied between 340 and 20 degrees, the average direction over that period should be near north (zero degrees), but taking the arithmetic average of 340 and 20 equals 180, or south. To produce a meaningful average of wind direction, we must use vector functions to decompose degrees into longitudinal and latitudinal dimensions, calculate the average (and/or a measure of variability), and recombine them to report the result in degrees.

Simply calculating the vector average of wind direction fails to take advantage of other information at our disposal, however. We will also take into account the wind speed, which acts as a weight on direction. For example, if the wind blows gently from the east for half of a day, and strongly from the north for half of the day, we would expect the average to be closer to north than to east.

Citation

This example and script was adapted from Grange, S.K. (2014). Technical note : Averaging wind speeds and directions, 1–12

Load necessary tools

library(knitr)
library(tidyverse)
## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.0 ──

## ✓ ggplot2 3.3.3     ✓ purrr   0.3.4
## ✓ tibble  3.1.0     ✓ dplyr   1.0.5
## ✓ tidyr   1.1.3     ✓ stringr 1.4.0
## ✓ readr   1.4.0     ✓ forcats 0.5.0

## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()

Create data to motivate example

Wind directions

We will be working with data obtained via API from Davis Instruments Weatherlink. The API returns wind direction data that are binned into 16 compass directions, and the compass directions are represented either as text or as an integer in [0, 15]. These integers must be translated into degree equivalents before other calculations can be made. We have confirmed with Davis Instruments that the integer values map to degrees in the order shown in the following code.

wind_dir <-
  data.frame(
    wdir_integer = c(0:15),
    wdir_text = c("N", "NNE", "NE", "ENE", "E", "ESE", "SE", "SSE", "S", "SSW", "SW", "WSW", "W", "WNW", "NW", "NNW"),
    wdir_degree = c(22.5 * c(0:15))
  )

Wind direction data translation

wind_dir %>% kable()
wdir_integer wdir_text wdir_degree
0 N 0.0
1 NNE 22.5
2 NE 45.0
3 ENE 67.5
4 E 90.0
5 ESE 112.5
6 SE 135.0
7 SSE 157.5
8 S 180.0
9 SSW 202.5
10 SW 225.0
11 WSW 247.5
12 W 270.0
13 WNW 292.5
14 NW 315.0
15 NNW 337.5

Example weather data

Data used in this example include 1000 rows, split into ten “days” with 100 values for each day. Wind speed is sampled from an integer vector and for this example is unitless. The wind direction data are sampled from the wind_dir data frame.

weather_data <-
  data.frame(
    day = rep(1:10, each = 100),
    wind_speed = sample(0:18, size = 1000, replace = TRUE),
    slice_sample(wind_dir, n = 1000, replace = TRUE)
  ) %>%
  glimpse()
## Rows: 1,000
## Columns: 5
## $ day          <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …
## $ wind_speed   <int> 18, 10, 1, 4, 2, 5, 13, 11, 4, 8, 8, 16, 7, 9, 1, 14, 3, …
## $ wdir_integer <int> 14, 5, 10, 7, 1, 7, 10, 15, 1, 5, 8, 8, 14, 2, 11, 13, 5,…
## $ wdir_text    <chr> "NW", "ESE", "SW", "SSE", "NNE", "SSE", "SW", "NNW", "NNE…
## $ wdir_degree  <dbl> 315.0, 112.5, 225.0, 157.5, 22.5, 157.5, 225.0, 337.5, 22…

Vector averaging and results

New variables are added here to the source data frame. The variables wdir_u and wdir_v comprise the latitudinal and longitudinal components of the wind direction vector.

In all summary functions, na.rm = TRUE is set because it is almost a given in real weather data that NA or NULL values will be reported.

## Calculate the u and v wind components, add as new variables in data frame
weather_data$wdir_u <- -weather_data$wind_speed * sin(2 * pi * weather_data$wdir_degree / 360)
weather_data$wdir_v <- -weather_data$wind_speed * cos(2 * pi * weather_data$wdir_degree / 360)
weather_data %>% glimpse()
## Rows: 1,000
## Columns: 7
## $ day          <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …
## $ wind_speed   <int> 18, 10, 1, 4, 2, 5, 13, 11, 4, 8, 8, 16, 7, 9, 1, 14, 3, …
## $ wdir_integer <int> 14, 5, 10, 7, 1, 7, 10, 15, 1, 5, 8, 8, 14, 2, 11, 13, 5,…
## $ wdir_text    <chr> "NW", "ESE", "SW", "SSE", "NNE", "SSE", "SW", "NNW", "NNE…
## $ wdir_degree  <dbl> 315.0, 112.5, 225.0, 157.5, 22.5, 157.5, 225.0, 337.5, 22…
## $ wdir_u       <dbl> 1.272792e+01, -9.238795e+00, 7.071068e-01, -1.530734e+00,…
## $ wdir_v       <dbl> -1.272792e+01, 3.826834e+00, 7.071068e-01, 3.695518e+00, …

## Calculate the average of wind vectors, grouped by time (in this case, "day")
weather_summary <-
  weather_data %>%
  group_by(day) %>%
  summarize(
    wind_speed_mean = mean(wind_speed, na.rm = TRUE),
    wdir_u_mean = mean(wdir_u, na.rm = TRUE),
    wdir_v_mean = mean(wdir_v, na.rm = TRUE),
    wdir_deg_avg = (atan2(wdir_u_mean, wdir_v_mean) * 360 / 2 / pi) + 180,
    .groups = "drop"
  )
weather_summary %>% kable()
day wind_speed_mean wdir_u_mean wdir_v_mean wdir_deg_avg
1 7.96 -0.0670350 0.6332023 173.95679
2 10.43 0.1480622 -0.3395205 336.43836
3 9.44 -0.7203740 0.1163207 99.17253
4 8.54 -0.8312698 -1.0027136 39.65942
5 7.90 0.5941624 0.0001448 269.98604
6 8.67 0.7982103 0.6269406 231.85275
7 9.12 1.0937407 -0.0001237 270.00648
8 9.14 -0.4085479 -0.1993275 63.99260
9 9.07 -0.2400419 0.9383821 165.65120
10 9.15 0.5248598 0.5170898 225.42726

It is possible to calculate the standard deviation of wind speed and wind direction, but implementing this will take additional work. Applying standard deviations to wind direction means that the y axis in figures will extend above 360 and below 0, which seems inappropriate. For wind speed, we also will be able to display max and min wind speed, if desired, alleviating the need for a measure of variability.

For information on the standard deviation of wind directions, see:

  • Wikipedia page on the Yamartino method
  • On the Algorithms Used to Compute the Standard Deviation of Wind Direction, Farrugia et al. 2009

Graphical display

This figure is laughably simple with only ten days included, but it’s representative of what wind speed and direction displays often look like.

weather_summary %>% 
  ggplot(aes(x = day, y = wind_speed_mean)) +
  geom_line(color = "red") +
  theme_bw()

weather_summary %>% 
  ggplot(aes(x = day, y = wdir_deg_avg)) +
  geom_line(color = "blue") +
  theme_bw()

Proof of concept

Let’s make sure that this works with wind directions that should average to north.

dir_vec <- c(310, 340, 355, 5, 15, 20)
sp_vec = sample(2:7)
u <- -sp_vec * sin(2 * pi * dir_vec / 360)
v <- -sp_vec * cos(2 * pi * dir_vec / 360)

The result should be close to 360 and not close to the average of dir_vec

## Result of vector averaging:
(atan2(mean(u), mean(v)) * 360 / 2 / pi) + 180
## [1] 357.1237

## Arithmetic mean of vector:
mean(dir_vec)
## [1] 174.1667

Code simplification

The code here could easily be streamlined into a unified block, but for implementation of this in our weather application, this script must be adapted to SQL, so for now it’s best to leave it like it is.