A condensed version of a paper entitled: “Violating Nyquist: Another Source of Significant Error in the Instrumental Temperature Record”.

By William Ward, 1/01/2019

The 4,900-word paper can be downloaded here: https://wattsupwiththat.com/wp-content/uploads/2019/01/Violating-Nyquist-Instrumental-Record-20190112-1Full.pdf

The 169-year long instrumental temperature record is built upon 2 measurements taken daily at each monitoring station, specifically the maximum temperature (Tmax) and the minimum temperature (Tmin). These daily readings are then averaged to calculate the daily mean temperature as Tmean = (Tmax+Tmin)/2. Tmax and Tmin measurements are also used to calculate monthly and yearly mean temperatures. These mean temperatures are then used to determine warming or cooling trends. This “historical method” of using daily measured Tmax and Tmin values for mean and trend calculations is still used today. However, air temperature is a signal and measurement of signals must comply with the mathematical laws of signal processing. The Nyquist-Shannon Sampling Theorem tells us that we must sample a signal at a rate that is at least 2x the highest frequency component of the signal. This is called the Nyquist Rate. Sampling at a rate less than this introduces aliasing error into our measurement. The slower our sample rate is compared to Nyquist, the greater the error will be in our mean temperature and trend calculations. The Nyquist Sampling Theorem is essential science to every field of technology in use today. Digital audio, digital video, industrial process control, medical instrumentation, flight control systems, digital communications, etc., all rely on the essential math and physics of Nyquist.

NOAA, in their USCRN (US Climate Reference Network) has determined that it is necessary to sample at 4,320-samples/day to practically implement Nyquist. 4,320-samples/day equates to 1-sample every 20 seconds. This is the practical Nyquist sample rate. NOAA averages these 20-second samples to 1-sample every 5 minutes or 288-samples/day. NOAA only publishes the 288-sample/day data (not the 4,320-samples/day data), so to align with NOAA the rate will be referred to as “288-samples/day” (or “5-minute samples”). (Unfortunately, NOAA creates naming confusion with their process of averaging down to a slower rate. It should be understood that the actual rate is 4,320-samples/day.) This rate can only be achieved by automated sampling with electronic instruments. Most of the instrumental record is comprised of readings of mercury max/min thermometers, taken long before automation was an option. Today, despite the availability of automation, the instrumental record still uses Tmax and Tmin (effectively 2-samples/day) instead of a Nyquist compliant sampling. The reason for this is to maintain compatibility with the older historical record. However, with only 2-samples/day the instrumental record is highly aliased. It will be shown in this paper that the historical method introduces significant error to mean temperatures and long-term temperature trends.

NOAA’s USCRN is a small network that was completed in 2008 and it contributes very little to the overall instrumental record. However, the USCRN data provides us a special opportunity to compare a high-quality version of the historical method to a Nyquist compliant method. The Tmax and Tmin values are obtained by finding the highest and lowest values among the 288 samples for the 24-hour period of interest.


NOAA USCRN Examples to Illustrate the Effect of Violating Nyquist on Mean Temperature

The following example will be used to illustrate how the amount of error in the mean temperature increases as the sample rate decreases. Figure 1 shows the temperature as measured at Cordova AK on Nov 11, 2017, using the NOAA USCRN 5-minute samples.


Figure 1: NOAA USCRN Data for Cordova, AK Nov 11, 2017

The blue line shows the 288 samples of temperature taken that day. It shows 24-hours of temperature data. The green line shows the correct and accurate daily mean temperature that is calculated by summing the value of each sample and then dividing the sum by the total number of samples. Temperature is not heat energy, but it is used as an approximation of heat energy. To that extent, the mean (green line) and the daily-signal (blue line) deliver the exact same amount of heat energy over the 24-hour period of the day. The correct mean is -3.3 °C. Tmax is represented by the orange line and Tmin by the grey line. These are obtained by finding the highest and lowest values among the 288 samples for the 24-hour period. The mean calculated from (Tmax+Tmin)/2 is shown by the red line. (Tmax+Tmin)/2 yields a mean of -4.7 °C, which is a 1.4 °C error compared to the correct mean.

Using the same signal and data from Figure 1, Figure 2 shows the calculated temperature means obtained from progressively decreased sample rates. These decreased sample rates can be obtained by dividing down the 288-sample/day sample rate by a factor of 4, 8, 12, 24, 48, 72 and 144. Therefore, the sample rates will correspond to: 72, 36, 24, 12, 6, 4 and 2-samples/day respectively. By properly discarding the samples using this method of dividing down, the net effect is the same as having sampled at the reduced rate originally. The corresponding aliasing that results from the lower sample rates, reveals itself as shown in the table in Figure 2.


Figure 2: Table Showing Increasing Mean Error with Decreasing Sample Rate

It is clear from the data in Figure 2, that as the sample rate decreases below Nyquist, the corresponding error introduced from aliasing increases. It is also clear that 2, 4, 6 or 12-samples/day produces a very inaccurate result. 24-samples/day (1-sample/hr) up to 72-samples/day (3-samples/hr) may or may not yield accurate results. It depends upon the spectral content of the signal being sampled. NOAA has decided upon 288-samples/day (4,320-samples/day before averaging) so that will be considered the current benchmark standard. Sampling below a rate of 288-samples/day will be (and should be) considered a violation of Nyquist.

It is interesting to point out that what is listed in the table as 2-samples/day yields 0.7 °C error. But (Tmax+Tmin)/2 is also technically 2-samples/day with an error of 1.4°C as shown in the table. How can this be possible? It is possible because (Tmax+Tmin)/2 is a special case of 2-samples per day because these samples are not spaced evenly in time. The maximum and minimum temperatures happen whenever they happen. When we sample properly, we sample according to a “clock” – where the samples happen regularly at exactly the same time of day. The fact that Tmax and Tmin happen at irregular times during the day causes its own kind of sampling error. It is beyond the scope of this paper to fully explain, but this error is related to what is called “clock jitter”. It is a known problem in the field of signal analysis and data acquisition. 2-samples/day, regularly timed, would likely produce better results than finding the maximum and minimum temperatures from any given day. The instrumental temperature record uses the absolute worst method of sampling possible – resulting in maximum error.

Figure 3 shows the same daily temperature signal as in Figure 1, represented by 288-samples/day (blue line). Also shown is the same daily temperature signal sampled with 12-samples/day (red line) and 4-samples/day (yellow line). From this figure, it is visually obvious that a lot of information from the original signal is lost by using only 12-samples/day, and even more is lost by going to 4-samples/day. This lost information is what causes the resulting mean to be incorrect. This figure graphically illustrates what we see in the corresponding table of Figure 2. Figure 3 explains the sampling error in the time-domain.


Figure 3: NOAA USCRN Data for Cordova, AK Nov 11, 2017: Decreased Detail from 12 and 4-Samples/Day Sample Rate – Time-Domain

Figure 4 shows the daily mean error between the USCRN 288-samples/day method and the historical method, as measured over 365 days at the Boulder CO station in 2017. Each data point is the error for that particular day in the record. We can see from Figure 4 that (Tmax+Tmin)/2 yields daily errors of up to ± 4 °C. Calculating mean temperature with 2-samples/day rarely yields the correct mean.


Figure 4: NOAA USCRN Data for Boulder CO – Daily Mean Error Over 365 Days (2017)

Let’s look at another example, similar to the one presented in Figure 1, but over a longer period of time. Figure 5 shows (in blue) the 288-samples/day signal from Spokane WA, from Jan 13 – Jan 22, 2008. Tmax (avg) and Tmin (avg) are shown in orange and grey respectively. The (Tmax+Tmin)/2 mean is shown in red (-6.9 °C) and the correct mean calculated from the 5-minute sampled data is shown in green (-6.2 °C). The (Tmax+Tmin)/2 mean has an error of 0.7 °C over the 10-day period.


Figure 5: NOAA USCRN Data for Spokane, WA – Jan13-22, 2008


The Effect of Violating Nyquist on Temperature Trends

Finally, we need to look at the impact of violating Nyquist on temperature trends. In Figure 6, a comparison is made between the linear temperature trends obtained from the historical and Nyquist compliant methods using NOAA USCRN data for Blackville SC, from Jan 2006 – Dec 2017. We see the trend derived from the historical method (orange line) starts approximately 0.2 °C warmer and has a 0.24 °C/decade warming bias compared to the Nyquist compliant method (blue line). Figure 7 shows the trend bias or error (°C/Decade) for 26 stations in the USCRN over a 7-12 year period. The 5-minute samples data gives us our reference trend. The trend bias is calculated by subtracting the reference from the (Tmaxavg+Tminavg)/2 derived trend. Almost every station exhibits a warming bias, with a few exhibiting a cooling bias. The largest warming bias is 0.24 °C/decade and the largest cooling bias is -0.17 °C/decade, with an average warming bias across all 26 stations of 0.06C. According to Wikipedia, the calculated global average warming trend for the period 1880-2012 is 0.064 ± 0.015 °C per decade. If we look at the more recent period that contains the controversial “Global Warming Pause”, then using data from Wikipedia, we get the following warming trends depending upon which year is selected for the starting point of the “pause”:

1996: 0.14°C/decade

1997: 0.07°C/decade

1998: 0.05°C/decade

While no conclusions can be made by comparing the trends over 7-12 years from 26 stations in the USCRN to the currently accepted long-term or short term global average trends, it can be instructive. It is clear that using the historical method to calculate trends yields a trend error and this error can be of a similar magnitude to the claimed trends. Therefore, it is reasonable to call into question the validity of the trends. There is no way to know for certain, as the bulk of the instrumental record does not have a properly sampled alternate record to compare it to. But it is a mathematical certainty that every mean temperature and derived trend in the record contains significant error if it was calculated with 2-samples/day.


Figure 6: NOAA USCRN Data for Blackville, SC – Jan 2006-Dec 2017 – Monthly Mean Trendlines


Figure 7: Trend Bias (°C/Decade) for 26 Stations in USCRN


1. Air temperature is a signal and therefore, it must be measured by sampling according to the mathematical laws governing signal processing. Sampling must be performed according to The Nyquist Shannon-Sampling Theorem.

2. The Nyquist-Shannon Sampling Theorem has been known for over 80 years and is essential science to every field of technology that involves signal processing. Violating Nyquist guarantees samples will be corrupted with aliasing error and the samples will not represent the signal being sampled. Aliasing cannot be corrected post-sampling.

3. The Nyquist-Shannon Sampling Theorem requires the sample rate to be greater than 2x the highest frequency component of the signal. Using automated electronic equipment and computers, NOAA USCRN samples at a rate of 4,320-samples/day (averaged to 288-samples/day) to practically apply Nyquist and avoid aliasing error.

4. The instrumental temperature record relies on the historical method of obtaining daily Tmax and Tmin values, essentially 2-samples/day. Therefore, the instrumental record violates the Nyquist-Shannon Sampling Theorem.

5. NOAA’s USCRN is a high-quality data acquisition network, capable of properly sampling a temperature signal. The USCRN is a small network that was completed in 2008 and it contributes very little to the overall instrumental record, however, the USCRN data provides us a special opportunity to compare analysis methods. A comparison can be made between temperature means and trends generated with Tmax and Tmin versus a properly sampled signal compliant with Nyquist.

6. Using a limited number of examples from the USCRN, it has been shown that using Tmax and Tmin as the source of data can yield the following error compared to a signal sampled according to Nyquist:

a. Mean error that varies station-to-station and day-to-day within a station.

b. Mean error that varies over time with a mathematical sign that may change (positive/negative).

c. Daily mean error that varies up to +/-4°C.

d. Long term trend error with a warming bias up to 0.24°C/decade and a cooling bias of up to 0.17°C/decade.

7. The full instrumental record does not have a properly sampled alternate record to use for comparison. More work is needed to determine if a theoretical upper limit can be calculated for mean and trend error resulting from use of the historical method.

8. The extent of the error observed with its associated uncertain magnitude and sign, call into question the scientific value of the instrumental record and the practice of using Tmax and Tmin to calculate mean values and long-term trends.

Reference section:

This USCRN data can be found at the following site: https://www.ncdc.noaa.gov/crn/qcdatasets.html

NOAA USCRN data for Figure 1 is obtained here:


NOAA USCRN data for Figure 4 is obtained here:


NOAA USCRN data for Figure 5 is obtained here:


NOAA USCRN data for Figure 6 is obtained here:


via Watts Up With That?


January 14, 2019 at 04:11PM

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