Daily Mean TemperatureEffectiveness of different
observational schemes
Colin Sowder, Lee Richardson, Alex Januzi, Dan Jones
General Outline
• Our objective was to assess the variance and bias corresponding to
different observational schemes for estimating mean temperature
• Our goal in looking at different observational schemes is to find the
most practical way of getting an accurate estimate for the daily mean.
• This gets more complicated the more you look into it, as different
locations and seasons follow different patterns.
• Because of this, we looked at observational schemes that could be
applied homogeneously to all stations
Daily Temperature Variations - Diurnal
Variation of Solar and Terrestrial Radiation
• the diurnal variation of incoming solar radiation
– it begins at sunrise
– it's a max at noon
– it shuts off at sunset
• the diurnal variation of earth-emitted terrestrial
radiation
– its trend is similar to the
– diurnal temperature trend:
– minimum at sunrise
– maximum at 3-5 PM
http://www-das.uwyo.edu/~zwang/atsc2000/Ch3.pdf
Daily Temperature Variations - Net Radiation
• The net radiation determines whether the surface
temperature rises, falls, or remains the same:
• net radiation = incoming solar - outgoing IR
• If the net radiation > 0, surface warms
(6 AM - 3-5 PM)
– If the net radiation < 0, surface cools
(3-5 PM - 6 AM)
• This explains why the warmest part of the year is in
July/August, not on 21 June during the summer
solstice.
http://www-das.uwyo.edu/~zwang/atsc2000/Ch3.pdf
•
The daily range of
temperature decreases as we
climb away from the earth’s
surface. There is less day-tonight variation in air
temperature at higher
altitudes compared to ground
level.
•
Typical temperature reading
stations are 2-3m off the
ground.
•
In summer, solar radiation is
more direct with higher net
radiation levels. This could
require more frequent
readings.
•
In winter, solar radiation is
more spread out. This could
require less frequent readings.
http://www-das.uwyo.edu/~zwang/atsc2000/Ch3.pdf
•
In mid-latitudes, solar radiation exhibits a pronounced annual maximum and minimum
and the seasonal difference in solar radiation is extreme in polar latitudes. As a result,
distinct winter-to-summer temperature contrasts are observed in middle and high
latitudes.
•
Our observed data (Visby and Iowa) falls within the 40 – 60 degree mid to high latitude
range, providing for similar effects of the spread of radiation with seasonal variations.
http://scijinks.nasa.gov/weather-v-climate
• Therefore, you could expect that in the summertime
with greater concentrations of solar radiation (more
net radiation, possibly greater lag times), the
temperatures would be more dynamic.
• Likewise in the winter, with solar radiation more
spread out, + and – net radiation levels would be
smaller, reducing the variability and need to record
temperature readings as often.
Month
January
July
Average
Sunrise
8:26 AM
4:00 AM
Average
Sunset
3:28 PM
9:45 PM
Source: TimeandDate.com
This is true in our Data as well, as the blue line (July Visby Data) is more variable than the
red line (January Visby Data).
The Rise and fall of the hourly mean temperature’s also correspond with the average
sunrise/sunset in each month.
Research Question
The questions we looked at were the effectiveness of the daily min and
max temperature as well as the effect of bias and variability of different
observational schemes.
We considered various hourly schemes with different number of
observations as well as different starting times.
The value that we were trying to predict was the mean temperature of
minute observations each day.
To answer these questions, we are using minute data from Visby, Sweden as well as
Red Oak, Iowa, because minute data is the finest resolution possible.
Commonly used methods of calculating daily mean
temperature in the literature:
• Mean of the 24 hourly means
• Mean of the maximum & minimum temperatures
• Weighted mean: 3 observations, last observation
weighted twice
• Mean of equally spaced observations (e.g. every 3
hours)
Source: Weiss, A., & Hays, C. J. (2005). Calculating daily
mean air temperatures by different methods: implications
from a non-linear algorithm. Agricultural and forest
meteorology, 128(1), 57-65.
Criteria for a good metric
• The Criteria we looked at for a good observational scheme were bias
and variance. The goal was a for both to be as small as possible
• To combine these two criteria, we used Mean Squared Error, where:
MSE = Var + (Bias)^2
Observational Schemes that we looked at
• We looked at estimates which recorded observations every hour, 2
hours, 3 hours (anything divisible by 24), etc.
• In each observational scheme, with the exception of the hourly, we
were able to “tune” our observations to start at different hours.
• Each observational scheme we looked at was evaluated with the MSE
in mind
• The MSE values in hour observational schemes were looked at in
comparison to the Daily min/max average that is used in many
countries.
Idea Behind Tuning Hourly Observations
• When taking hourly estimates, we explored starting our estimates at
different times, rather than at the default 0 mark.
• The less observations for certain hourly schemes, the more options
for tuning there were, ex: you could start 3 hourly observations at the
0, 1, or 2 hour
-8
-7
-6
-5
-4
Mean Temperature
-3
-2
-1
Visby-Daily Mean
0
5
10
15
Day
20
25
30
-4
-5
-6
-7
-8
Mean Temperature
-3
-2
-1
Visby-Daily Range
0
5
10
15
Day
20
25
30
-4
-5
-6
-7
-8
Mean Temperature
-3
-2
-1
Visby-Hourly Mean
0
5
10
15
Day
20
25
30
-4
-5
-6
-7
-8
Mean Temperature
-3
-2
-1
Visby-2 Hourly Mean
0
5
10
15
Day
20
25
30
-4
-5
-6
-7
-8
Mean Temperature
-3
-2
-1
Visby-January 2 Hourly Mean Tuned
0
5
10
15
Day
20
25
30
-4
-5
-6
-7
-8
Mean Temperature
-3
-2
-1
Visby-3 Hourly Mean
0
5
10
15
Day
20
25
30
-4
-5
-6
-7
-8
Mean Temperature
-3
-2
-1
Visby-January 3 Hourly Mean Tuned
0
5
10
15
Day
20
25
30
-4
-5
-6
-7
-8
Mean Temperature
-3
-2
-1
Visby-4 Hourly Mean
0
5
10
15
Day
20
25
30
-4
-5
-6
-7
-8
Mean Temperature
-3
-2
-1
Visby-January 4 Hourly Mean Tuned
0
5
10
15
Day
20
25
30
-4
-5
-6
-7
-8
Mean Temperature
-3
-2
-1
Visby-6 Hourly Mean
0
5
10
15
Day
20
25
30
-4
-5
-6
-7
-8
Mean Temperature
-3
-2
-1
Visby-January 6 Hourly Mean Tuned
0
5
10
15
Day
20
25
30
-4
-5
-6
-7
-8
Mean Temperature
-3
-2
-1
Visby-8 Hourly Mean
0
5
10
15
Day
20
25
30
-4
-5
-6
-7
-8
Mean Temperature
-3
-2
-1
Visby-January 8 Hourly Mean Tuned
0
5
10
15
Day
20
25
30
-4
-5
-6
-7
-8
Mean Temperature
-3
-2
-1
Visby-12 Hourly Mean
0
5
10
15
Day
20
25
30
-4
-5
-6
-7
-8
Mean Temperature
-3
-2
-1
Visby-January 12 Hourly Mean Tuned
0
5
10
15
Day
20
25
30
Visby - January
Metric First Obs
Range
Hourly
0:00
2-hourly
0:00
3-hourly
0:00
4-hourly
0:00
6-hourly
0:00
8-hourly
0:00
12-hourly
0:00
Bias Variance MSE
-0.093 0.181 0.184
0.000 0.004 0.003
0.017 0.012 0.012
-0.011 0.030 0.029
0.016 0.056 0.054
0.010 0.116 0.112
-0.139 0.349 0.357
0.284 0.454 0.521
Visby – January Tuned
Metric First Obs
Range
Hourly
0:00
2-hourly
1:00
3-hourly
2:00
4-hourly
2:00
6-hourly
3:00
8-hourly
6:00
12-hourly
4:00
Bias Variance MSE
-0.093 0.181 0.184
0.000 0.004 0.003
-0.018 0.003 0.003
0.000 0.005 0.005
0.018 0.010 0.010
-0.031 0.015 0.016
-0.033 0.082 0.080
-0.054 0.123 0.122
-1
Range
Hourly
2 Hourly
3 Hourly
4 Hourly
6 Hourly
8 Hourly
12 Hourly
-4
-5
-6
0.0
-8
-7
-0.5
-1.0
Difference from mean
0.5
Mean Temperature
-3
-2
Visby-January Differences
0
5
10
15
20
25
25
30
Day
0
5
10
15
Day
20
30
-1
Range
Hourly
2 Hourly
3 Hourly
4 Hourly
6 Hourly
8 Hourly
12 Hourly
-4
-5
-6
0.0
-8
-7
-0.5
-1.0
Difference from mean
0.5
Mean Temperature
-3
-2
Visby-January Differences Tuned
0
5
10
15
20
25
25
30
Day
0
5
10
15
Day
20
30
-1
Range
Hourly
2 Hourly
3 Hourly
4 Hourly
6 Hourly
8 Hourly
12 Hourly
-4
-5
-6
22
-7
20
-8
18
0
5
10
16
Mean Temperature
24
Mean Temperature
-3
26
-2
Visby-July Mean Temperatures
15
20
25
25
30
Day
0
5
10
15
Day
20
30
-1
Range
Hourly
2 Hourly
3 Hourly
4 Hourly
6 Hourly
8 Hourly
12 Hourly
-4
-5
-6
-7
0
-8
-1
0
5
10
15
20
25
25
30
Day
-2
Difference from Mean
1
Mean Temperature
-3
2
-2
Visby-July Differences
0
5
10
15
Day
20
30
Visby – July
Metric First Obs
Range
Hourly
0:00
2-hourly
1:00
3-hourly
2:00
4-hourly
2:00
6-hourly
3:00
8-hourly
6:00
12-hourly
4:00
Bias Variance MSE
0.162 0.521 0.531
-0.002 0.006 0.005
-0.025 0.016 0.016
-0.017 0.027 0.027
-0.009 0.071 0.069
-0.305 0.276 0.360
0.080 0.348 0.343
0.133 0.384 0.389
-1
-4
-5
-8
-7
75
-6
80
Mean Temperature
-3
85
-2
Read Oak-June Mean Temperature
0
70
Mean Temperature
Range
Hourly
2 Hourly
3 Hourly
4 Hourly
6 Hourly
8 Hourly
12 Hourly
5
10
15
20
25
25
30
Day
0
5
10
15
Day
20
30
-1
Range
Hourly
2 Hourly
3 Hourly
4 Hourly
6 Hourly
8 Hourly
12 Hourly
-4
-5
-6
-7
0
-8
-2
0
5
10
15
20
25
25
30
Day
-4
Difference from Mean
2
Mean Temperature
-3
4
-2
Read Oak-June Differences
0
5
10
15
Day
20
30
Red Oak - June
Metric First Obs
Range
Hourly
0
2-hourly
1
3-hourly
2
4-hourly
2
6-hourly
3
8-hourly
6
12-hourly
4
Bias Variance MSE
-0.381 2.150 2.225
-0.006 0.043 0.041
-0.010 0.028 0.027
0.119 0.077 0.088
-0.160 0.158 0.178
0.111 0.111 0.120
-1.031 0.816 1.853
0.813 2.022 2.618
A return to Visby – One more thing
• Recall the MSE for daily range in July was 0.531
• The original calculations for Visby produced the following table:
Metric First Obs
Range
Hourly
0:00
2-hourly
1:00
3-hourly
2:00
4-hourly
2:00
6-hourly
3:00
8-hourly
6:00
12-hourly
4:00
• What happened?
Bias Variance MSE
-0.057 1.881 1.824
-0.001 0.006 0.005
-0.025 0.016 0.016
-0.016 0.027 0.027
-0.009 0.071 0.069
-0.305 0.276 0.360
0.080 0.348 0.343
0.134 0.382 0.388
Visby – July 7th
15
10
5
Temperature
20
Visby Temperature July 7
0
200
400
600
800
1000
1200
1400
Minute
In addition to missing data, there was an outlier of 2.6 C
Visby – July Corrected
• Eliminating the outlier gives the previous results:
Metric First Obs
Range
Hourly
0:00
2-hourly
1:00
3-hourly
2:00
4-hourly
2:00
6-hourly
3:00
8-hourly
6:00
12-hourly
4:00
Bias Variance MSE MSE(old)
0.162 0.521 0.531 1.824
-0.002 0.006 0.005
0.005
-0.025 0.016 0.016
0.016
-0.017 0.027 0.027
0.027
-0.009 0.071 0.069
0.069
-0.305 0.276 0.360
0.360
0.080 0.348 0.343
0.343
0.133 0.384 0.389
0.388
• Results change dramatically for Range-based metric.
• Maxes and Mins may be more likely to be in error.
Conclusions
• More observations per day are better!
• The mean of max and min may not be the best measure of daily mean
temperature.
• Two evenly spaced daily observations outperformed by MSE for two of the three
study sites (Red Oak)
• Susceptible to errors in measurement
• “Tuning” improves Mean Square Error
• May need to be site specific
• Winter temperatures have lower Mean Square Error than Summer
temperatures at the same site
• Fewer observations needed per day to have the same MSE
Further Questions
• What is the best balance of MSE to number of observations per day?
• What can practically be stored?
• Can we use our knowledge of temperature to create even better
metrics?
• For example: Tie observation times to the Sun
• Can the tuning parameter be set using this knowledge?
• How robust are the various metrics to missing data?
• What effect does measurement error have on these metrics?