# Introduction

One of the best sources of weather data in the United States comes from the National Weather Service's Cooperative Observer Network (COOP), which is available from NCDC. It's daily data, collected by volunteers at more than 10,000 locations. We participate in this program at our house (station id DW1454 / GHCND:USC00503368), collecting daily minimum and maximum temperature, liquid precipitation, snowfall and snow depth. We also collect river heights for Goldstream Creek as part of the Alaska Pacific River Forecast Center (station GSCA2). Traditionally, daily temperature measurements were collecting using a minimum maximum thermometer, which meant that the only way to calculate average daily temperature was by averaging the minimum and maximum temperature. Even though COOP observers typically have an electronic instrument that could calculate average daily temperature from continuous observations, the daily minimum and maximum data is still what is reported.

In an earlier post we looked at methods used to calculate average daily temperature, and if there are any biases present in the way the National Weather Service calculates this using the average of the minimum and maximum daily temperature. We looked at five years of data collected at my house every five minutes, comparing the average of these temperatures against the average of the daily minimum and maximum. Here, we will be repeating this analysis using data from the Climate Reference Network stations in the United States.

The US Climate Reference Network is a collection of 132 weather stations that are properly sited, maintained, and include multiple redundant measures of temperature and precipitation. Data is available from http://www1.ncdc.noaa.gov/pub/data/uscrn/products/ and includes monthly, daily, and hourly statistics, and sub-hourly (5-minute) observations. We’ll be focusing on the sub-hourly data, since it closely matches the data collected at my weather station.

A similar analysis using daily and hourly CRN data appears here.

# Getting the raw data

I downloaded all the data using the following Unix commands:

```
$ wget http://www1.ncdc.noaa.gov/pub/data/uscrn/products/stations.tsv
$ wget -np -m http://www1.ncdc.noaa.gov/pub/data/uscrn/products/subhourly01/
$ find www1.ncdc.noaa.gov/ -type f -name 'CRN*.txt' -exec gzip {} \;
```

The code to insert all of this data into a database can be found
here.
Once inserted, I have a table named `crn_stations` that has the
station data, and one named `crn_subhourly` with the five minute
observation data.

# Methods

Once again, we’ll use R to read the data, process it, and produce plots.

## Libraries

Load the libraries we need:

```
library(dplyr)
library(lubridate)
library(ggplot2)
library(scales)
library(grid)
```

Connect to the database and load the data tables.

```
noaa_db <- src_postgres(dbname="noaa", host="mason")
crn_stations <- tbl(noaa_db, "crn_stations") %>%
collect()
crn_subhourly <- tbl(noaa_db, "crn_subhourly")
```

Remove observations without temperature data, group by station and date, calculate average daily temperature using the two methods, remove any daily data without a full set of data, and collect the results into an R data frame. This looks very similar to the code used to analyze the data from my weather station.

```
crn_daily <-
crn_subhourly %>%
filter(!is.na(air_temperature)) %>%
mutate(date=date(timestamp)) %>%
group_by(wbanno, date) %>%
summarize(t_mean=mean(air_temperature),
t_minmax_avg=(min(air_temperature)+
max(air_temperature))/2.0,
n=n()) %>%
filter(n==24*12) %>%
mutate(anomaly=t_minmax_avg-t_mean) %>%
select(wbanno, date, t_mean, t_minmax_avg, anomaly) %>%
collect()
```

The two types of daily average temperatures are calculated in this step:

```
summarize(t_mean=mean(air_temperature),
t_minmax_avg=(min(air_temperature)+
max(air_temperature))/2.0)
```

Where `t_mean` is the value calculated from all 288 five minute
observations, and `t_minmax_avg` is the value from the daily minimum
and maximum.

Now we join the observation data with the station data. This attaches station information such as the name and latitude of the station to each record.

```
crn_daily_stations <-
crn_daily %>%
inner_join(crn_stations, by="wbanno") %>%
select(wbanno, date, state, location, latitude, longitude,
t_mean, t_minmax_avg, anomaly)
```

Finally, save the data so we don’t have to do these steps again.

```
save(crn_daily_stations, file="crn_daily_averages.rdata")
```

# Results

Here are the overall results of the analysis.

```
summary(crn_daily_stations$anomaly)
```

## Min. 1st Qu. Median Mean 3rd Qu. Max. ## -11.9000 -0.1028 0.4441 0.4641 1.0190 10.7900

The average anomaly across all stations and all dates is 0.44 degrees Celsius (0.79 degrees Farenheit). That’s a pretty significant error. Half the data is between −0.1 and 1.0°C (−0.23 and +1.8°F) and the full range is −11.9 to +10.8°C (−21.4 to +19.4°F).

# Plots

Let’s look at some plots.

## Raw data by latitude

To start, we’ll look at all the anomalies by station latitude. The plot only shows one percent of the actual anomalies because plotting 512,460 points would take a long time and the general pattern is clear from the reduced data set.

```
set.seed(43)
p <- ggplot(data=crn_daily_stations %>% sample_frac(0.01),
aes(x=latitude, y=anomaly)) +
geom_point(position="jitter", alpha="0.2") +
geom_smooth(method="lm", se=FALSE) +
theme_bw() +
scale_x_continuous(name="Station latitude", breaks=pretty_breaks(n=10)) +
scale_y_continuous(name="Temperature anomaly (degrees C)",
breaks=pretty_breaks(n=10))
print(p)
```

The clouds of points show the differences between the min/max daily average and the actual daily average temperature, where numbers above zero represent cases where the min/max calculation overestimates daily average temperature. The blue line is the fit of a linear model relating latitude with temperature anomaly. We can see that the anomaly is always positive, averaging around half a degree at lower latitudes and drops somewhat as we proceed northward. You also get a sense from the actual data of how variable the anomaly is, and at what latitudes most of the stations are found.

Here are the regression results:

```
summary(lm(anomaly ~ latitude, data=crn_daily_stations))
```

## ## Call: ## lm(formula = anomaly ~ latitude, data = crn_daily_stations) ## ## Residuals: ## Min 1Q Median 3Q Max ## -12.3738 -0.5625 -0.0199 0.5499 10.3485 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 0.7403021 0.0070381 105.19 <2e-16 *** ## latitude -0.0071276 0.0001783 -39.98 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.9632 on 512458 degrees of freedom ## Multiple R-squared: 0.00311, Adjusted R-squared: 0.003108 ## F-statistic: 1599 on 1 and 512458 DF, p-value: < 2.2e-16

The overall model and coefficients are highly significant, and show a slight decrease in the positive anomaly as we move farther north. Perhaps this is part of the reason why the analysis of my station (at a latitude of 64.89) showed an average anomaly close to zero (−0.07°C / −0.13°F).

## Anomalies by month and latitude

One of the results of our earlier analysis was a seasonal pattern in the anomalies at our station. Since we also know there is a latitudinal pattern, in the data, let’s combine the two, plotting anomaly by month, and faceting by latitude.

Station latitude are binned into groups for plotting, and the plots themselves show the range that cover half of all anomalies for that latitude category × month. Including the full range of anomalies in each group tends to obscure the overall pattern, and the plot of the raw data didn’t show an obvious skew to the rarer anomalies.

Here’s how we set up the data frames for the plot.

```
crn_daily_by_month <-
crn_daily_stations %>%
mutate(month=month(date),
lat_bin=factor(ifelse(latitude<30, '<30',
ifelse(latitude>60, '>60',
paste(floor(latitude/10)*10,
(floor(latitude/10)+1)*10,
sep='-'))),
levels=c('<30', '30-40', '40-50',
'50-60', '>60')))
summary_stats <- function(l) {
s <- summary(l)
data.frame(min=s['Min.'],
first=s['1st Qu.'],
median=s['Median'],
mean=s['Mean'],
third=s['3rd Qu.'],
max=s['Max.'])
}
crn_by_month_lat_bin <-
crn_daily_by_month %>%
group_by(month, lat_bin) %>%
do(summary_stats(.$anomaly)) %>%
ungroup()
station_years <-
crn_daily_by_month %>%
mutate(year=year(date)) %>%
group_by(wbanno, lat_bin) %>%
summarize() %>%
group_by(lat_bin) %>%
summarize(station_years=n())
```

And the plot itself. At the end, we’re using a function called
`facet_adjust`, which adds x-axis tick labels to the facet on the
right that wouldn't ordinarily have them. The code comes from this
stack overflow
post.

```
p <- ggplot(data=crn_by_month_lat_bin,
aes(x=month, ymin=first, ymax=third, y=mean)) +
geom_hline(yintercept=0, alpha=0.2) +
geom_hline(data=crn_by_month_lat_bin %>%
group_by(lat_bin) %>%
summarize(mean=mean(mean)),
aes(yintercept=mean), colour="darkorange", alpha=0.5) +
geom_pointrange() +
facet_wrap(~ lat_bin, ncol=3) +
geom_text(data=station_years, size=4,
aes(x=2.25, y=-0.5, ymin=0, ymax=0,
label=paste('n =', station_years))) +
scale_y_continuous(name="Range including 50% of temperature anomalies") +
scale_x_discrete(breaks=1:12,
labels=c('Jan', 'Feb', 'Mar', 'Apr', 'May', 'Jun',
'Jul', 'Aug', 'Sep', 'Oct', 'Nov', 'Dec')) +
theme_bw() +
theme(axis.text.x=element_text(angle=45, hjust=1, vjust=1.25),
axis.title.x=element_blank())
facet_adjust(p)
```

Each plot shows the range of anomalies from the first to the third quartile (50% of the observed anomalies) by month, with the dot near the middle of the line at the mean anomaly. The orange horizontal line shows the overall mean anomaly for that latitude category, and the count at the bottom of the plot indicates the number of “station years” for that latitude category.

It’s clear that there are seasonal patterns in the differences between the mean daily temperature and the min/max estimate. But each plot looks so different from the next that it’s not clear if the patterns we are seeing in each latitude category are real or artificial. It is also problematic that three of our latitude categories have very little data compared with the other two. It may be worth performing this analysis in a few years when the lower and higher latitude stations have a bit more data.

# Conclusion

This analysis shows that there is a clear bias in using the average of minimum and maximum daily temperature to estimate average daily temperature. Across all of the CRN stations, the min/max estimator overestimates daily average temperature by almost a half a degree Celsius (0.8°F).

We also found that this error is larger at lower latitudes, and that there are seasonal patterns to the anomalies, although the seasonal patterns don’t seem to have clear transitions moving from lower to higher latitudes.

The current length of the CRN record is quite short, especially for the sub-hourly data used here, so the patterns may not be representative of the true situation.