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The observed basic weather variables are the main representative of climate trends and the atmosphere. The unresolved meteorological scale in weather observation such as micro scale, can produce a noticeable bias in amplitude, frequency, phase and climate trend of each observed variable time series. The bias in climate trend due to a small scale eddy can be as high as the amplitude of the eddy which could be greater than 1 ° C in a temperature trend. Such biased measurements of the state of the atmosphere limit all climate related studies.

The earth and its surrounding atmosphere change constantly. The changes in our atmosphere are mainly (considering urbanization, land use change and atmospheric chemistry as noise relative to the rest of solar system harmonics) driven by forces initiated by the earth’s position and movement in the solar system and the resulting incoming solar radiation. The incoming solar energy to the earth system varies with a few harmonics. These harmonics are not limited by but originated from the following facts [

The simple yet remarkable famous question “Does wind have a speed?” was raised by Lewis Fry Richardson due to the difficulty that he and other scientists of the time had in measuring the horizontal divergence with sufficient accuracy. Richardson performed a numerical integration of the governing equations of motion, which had been suggested by Bjerknes earlier, over a horizontal grid of about 200 km and four vertical layers of approximately 200 hPa, centered over Germany. He used the observation made at 7 UTC (Universal Coordinate Time) on May 20^{th}, 1910. He computed the time derivative of the pressure in central Germany between 4 and 10 UTC. Although the predicted 6-h pressure changes had huge discouraging errors [

It becomes obvious today that the inadequacies of observation alone would have condemned any attempt, of which Richardson was well aware. Besides, we know today, the fastest traveling signal (sound waves) in his experiment travels at about 300 m/s and the integration needed to be done faster in a smaller time increment (about 10 minutes). The experiment failed due mostly to the fact that the initial conditions were not balanced [

Atmospheric scientists know that when a front is involved the scale is about hours to days in time and 100 km to 1000 km in space (meso-α scale). When the heat exchange via turbulence is relevant the scale of space changes to 1 mm to 1 meter and time to seconds (micro-σ scale).

The disconnection between the data and natural solar system frequencies can be partially explained by the short length of available data and biases in our measurements. Weather observation and weather practice evolved based on instrument availability and resources more than on classical sampling theory. The objective of this paper is to explain the climate uncertainty that is caused by high frequency turbulence in the atmosphere. How do the biases cause some predictions to fail and some not. How do the sampling frequencies and down sampling (such as monthly mean) add bias in our observations and compromise some climate trends.

The Nyquist-Shannon [_{0}, then it can be collected with all the information in the signal by sampling at discrete times, as long as the sample rate is greater than 2f_{0}. Unfortunately, while the theorem is simple and straightforward, it can be very misleading when one tries to apply it to the atmosphere. One of the challenges is the fact that in nature we rarely have a perfect band-limited signal and aliasing (an artifact caused when different signals become indistinguishable) occurs. To avoid aliasing, we need to sample the signal faster than the highest frequencies or at least filter (low pass filter) the signal prior to sampling to remove the highest frequencies as much as we can, although filtering might change the phase and amplitude of a signal.

In designing sample-time or sample-space systems, the variables that we need to consider are signal accuracy (phase, amplitude and frequencies are equally important in atmospheric measurements) and various kinds of system cost (power consumption, dollar cost, etc.). Increasing the signal sample rate will always increase the signal accuracy. To sample the atmosphere based on the Nyquist-Shannon sampling theorem we need to record weather observations faster than every second in time and at least every 50 m in space, if we assume the band limits of atmospheric signals are about micro-α (

As there is no access to long time high frequency observations and to the real state of the atmosphere, we used a simple sinusoid wave as representative of a harmonic atmosphere. We considered that air temperature (2 m air temperature for example) at one station is a composition of 7 sinusoid harmonic signals with equal phase and wave numbers as presented by Equations (1) to (3). The wave number ( K ) is equal to 1 and the phase ( φ ) is zero for all harmonics. These seven harmonics are representative of turbulence, hourly, daily, monthly and annual temperature changes. A day is considered to be 23.9 hours and a month 29.5 days while a year is 365.25 days (as explained in the introduction).

Signal ( t ) = ∑ i = 1 7 ( amplitude ( i ) × s i n ( 2 π K frequencies ( i ) × t + φ ( i ) ) ) (1)

amplitude = [ 3,1,1.5,4,6,18 ] (2)

Periods = [ 2.5 seconds ,1.0001 minutes ,30 minutes ,0.5 day ,1 day ,1 month ,1 year ] (3)

Then the simplified temperature signal is sampled every 5 seconds and averaged every 720 samples to present the hourly temperature in the same fashion as Canadian data loggers sample the air temperature at automated stations. The five second sampling is common among Canadian automated stations and many other weather stations. The C-1 climate station in Niwot Ridge, Colorado, USA records the peak wind gust information instantaneously with a sampling interval of 5 seconds, and the other measurements are one hour averages comprised of 720 samples collected at 5 second intervals. At North Dakota Agriculture Weather Network (NDAWN) wind speed and direction are measured every 5 seconds and are averaged hourly and daily. Air temperature, relative humidity, solar radiation, bare soil temperature, and turf soil temperature are measured every 60 seconds and are averaged hourly. The hourly artificial temperature from the sampling is compared with the original signal (^{−}^{11} per hour (

We also tried to calculate the daily and monthly average based on a calendar when days are 24 hours and a year is 365 days. Then we compared the trend of monthly average with the 5 second sampling and 1 second sampling trends.

The above example is a simplified case with just 7 harmonics and no noise

Normal frequency (f_{0}/f_{s}) | Bandwidth (f_{0}) | Average annual bias | Annual trend difference |
---|---|---|---|

0.04 0.3 0.4 0.5 0.6 1 2 3 4 5 | 0.008 0.06 0.08 0.1 0.12 0.2 0.4 0.6 0.8 1 | −0.0008 1.1038 1.4470 1.7628 2.05 −0.0006 −0.0006 −0.0005 −0.0005 −0.0005 | −0.0015 to −0.0089 −0.0015 to −0.0089 −0.0015 to −0.0089 −0.0015 to −0.0089 −0.0015 to −0.0089 −2.8546 to −2.8443 −1.7648 to −1.7544 +1.7619 to +1.7723 +2.8521 to +2.8621 −0.0015 to −0.0089 |

while the real atmosphere is a continuous spectrum of harmonics and variation of noises such as industrial pollution and urbanization. The distribution of atmospheric measurements in space and time is far behind any sampling theory and meanwhile archiving and down sampling are based on calendar frequencies rather than natural frequencies.

Atmospheric measurements and atmospheric science constantly fall behind due to the limitation and quality of data. There is no doubt that increasing the number of observations in horizontal space and time is the ultimate solution. Meanwhile the sampling can be improved by considering: 1) magnitude and frequencies of local unresolved scale of motions (higher frequencies than sampling rate turbulence); and 2) down sampling based on real existing frequencies rather than the conventionally agreed frequencies. Considering the direct relation of scales in space and time of a phenomenon of interest and limitation of our available data to resolve the long term trend of such a phenomenon is crucial. In cases where the climate data cannot resolve the spatial = temporal scale, we use NWP models and data assimilation or available analysis. The combination of knowledge of mechanism and data through high resolution analyses can be a temporary alternative.

The observed weather variables are used in many fields including biology, civil engineering, hydrology, ecology, finance and meteorology. Although we learned from Lorenz [

The research was supported by a Natural Science and Engineering Research Council of Canada Discovery Grant to Edward A. Johnson. Many thanks to Prof. Kiyoko Miyanishi for her comments.

Bakhshaii, A. and Johnson, E.A. (2017) Atmospheric Observation under Sampling Problem: The Impact of Unresolved Micro-Scale Boundary Layer Eddies on Climate Trends. Open Journal of Statistics, 7, 964-971. https://doi.org/10.4236/ojs.2017.76068