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TOBS And Margins Of Error

July 5, 2014

By Paul Homewood



Reader Phil left the following comment about TOBS adjustments, which warrants its own post.

He suggests that the margin of error associated with TOBS variability is so large as to make any adjustment worthless for estimating temperature trends.




Although I have not done an exhaustive review of the literature, only TOBS error means appear to have been studied/modeled/adjusted. No one, it seems, has ever looked at TOBS error variances, at least to my knowledge. While an adjustment of the mean may seem a logical way to correct for biases, it does not correct nor have any effect on variances. If the TOBS error introduces enough uncertainty, then a trend may not be statistically significant, even if there is a bias correction.

I have spent several years looking at TOBS in the hope to someday publish something, but, for several reasons, it doesn’t look like I will be able to do so by myself at this time. This chart ( is an example of some of my research. It shows the “Confidence Interval” due to TOBS adjustments. This is calculated as follows.

Using hourly data, I calculated an “actual mean” as of midnight each day. Although there are several ways to approximate the “actual mean” in this manner (I looked at the rectangular approximation, the trapezoidal approximation and Simpson’s Rule), the differences are not very large. For simplicity, I used the rectangular approximation (i.e. adding the 24 hourly temperatures and dividing by 24). Then, I computed a maximum and a minimum for the previous 24 hrs, using the highest hourly number as the maximum and the lowest hourly number as the minimum.* I then repeated this for each “time of observation”, by sliding the 24 hour window one hour at a time for each of the 24 hours. A TOBS “error” was then calculated for each hour by taking the difference between the maxmin mean and the “actual mean” for the 24 different “times of observation.” I then calculated the standard deviation of the errors for each hour of each day over a one year period. The chart below was calculated using hourly data for Fort Smith Municipal Airport, Arkansas for 1984. It appears to be typical.

The “confidence interval” is calculated by taking the standard deviation of the temperature in Fahrenheit, converting it to Celsius, multiplying by 1.96 (2 sigmas) and then dividing by two to obtain a plus and minus “X” pseudo confidence interval. This purports to show the variance of the TOBS error due purely to a change in the time of observation using actual data.

The implication of this seem obvious. Since the rate of warming is supposedly about 1°C per century, it would seem that any trend based on afternoon observations, where the TOBS error appears to create an additional ±2.5°C uncertainty (for 5:00 P.M.), would not be statistically significant. Maybe a claim can be made that this uncertainty is reduced by appealing to various statistical miracles, but, at first sight, this would seem to be a significant hurdle to overcome.

Preliminarily, I would submit, therefore, that no conclusions regarding long-term trends can reasonably be made based on stations with afternoon observation times. Conclusions regarding long-term trends based on stations with morning observation times have a much smaller TOBS uncertainty (±1.5°C for 7:00 A.M.), but this is still large. I would say that this analysis of the apparently large uncertainties created by TOBS errors cannot be used either to prove or disprove (C)AGW or any other theory. Instead, it would appear to highlight serious problems with the available data that preclude any strong conclusions. Likewise, I think this analysis calls into question the validity of any TOBS adjustments.

It would appear that there is enough variability in the shape of the daily temperature curve that an adjustment of the mean cannot produce a bias correction within an uncertainty that is useful for estimating temperature trends. I would also say that, although TOBS corrections are not done for all global data, the TOBS error uncertainty shown in this chart is probably present in most global data, as probably relatively few stations have an effective observation time of midnight.

Since I still have some hope to publish this someday, I hereby claim copyright, and everything else I can to preserve whatever I can for as long as I can.

*Karl et al., 1986 ( has a discussion beginning on page 7 of the pdf (pg 151 of the published paper) of the differences between the actual maximum and minimum and the hourly max and min. Please refer to Table 3 in that publication for specifics, but these differences appear to be an order of magnitude smaller than the calculated TOBS error uncertainties.

  1. RossG permalink
    July 5, 2014 3:23 pm

    I suppose one way to test the impact of TOBS is to change the ‘standard’ time of day for the adjustment from midnight to noon. Then rerun everything and see if the anomalies change. Conceptually a switch from a cold time of day (midnight) to a warm time of day should reverese a bias caused by the TOBS algorithim.

  2. ChrisM permalink
    July 5, 2014 6:52 pm

    I had concerns about TOBS too. Fortunately, I work on a power station so had access to good quality temperature data. By getting the hourly max and mins over a 15 month period, I could then determine the effect of TOBS and if it was stable. It is real and a consistent offset, but only on an annual basis. If the analysis is on monthly data, then the offset variability is significant.

  3. Phil permalink
    July 6, 2014 9:10 am


    Thanks for your thoughts. Let me try to take things one sentence at a time.

    QUOTE: “I suppose one way to test the impact of TOBS is to change the ‘standard’ time of day for the adjustment from midnight to noon.”

    It is my understanding that the meteorological day, if you will, is defined to be the same as the calendar day – that is, from midnight to midnight. Historically, few stations had an observation time of midnight. The TOBS error or time of observation error is the difference between the average of the maximum temperature since the maxmin thermometers were reset and the minimum temperature since they were reset at the time of observation compared to a hypothetical average temperature if these thermometers were observed and reset at midnight of each day. It is this hypothetical reading that is being estimated by applying a TOBS adjustment to the actual measurements.

    It is important to understand that the hypothetical thermometer observation at midnight of a station many decades ago is not known. In the early part of the 20th Century most US COOP stations had observation times of 5:00 P.M., IIRC. During the middle part of the 20th Century these observation times were changed to mostly 7:00 A.M. Few, if any, stations had observation times of midnight. Today temperature is mostly measured electronically and more continuously so there is no time of observation error. The data that I used was measured hourly.

    Changing the definition of the standard day from midnight to midnight to noon to noon would cause an ambiguity as to which day the temperature refers to. Changing the standard would also change the TOBS error, but I don’t think it would test the “impact of TOBs,” because the observation times for data collected a long time ago don’t change.

    QUOTE: “Then rerun everything and see if the anomalies change.”

    At this point anomalies are not used.

    QUOTE: “Conceptually a switch from a cold time of day (midnight) to a warm time of day should reverese a bias caused by the TOBS algorithim.”

    If you are referring to a bias caused by the TOBS adjustment algorithm, then I would say no. If the bias of the TOBS adjustment algorithm is assumed to be about equal to the TOBS error, my analysis deals with the variance of the error and not with the bias. It seems that nobody has looked at the variance of the error and therefore at the uncertainty of the TOBS adjustments. In fact, it seems as if NO uncertainty at all is attributed to the adjustments, when my analysis appears to indicate that the uncertainty of the TOBS adjustments may indeed be very large, especially for the stations with afternoon observation times.

  4. Phil permalink
    July 6, 2014 9:13 am


    I would be interested in your analysis if you would be so kind as to provide specifics. It is difficult to reply absent specifics.

  5. RossG permalink
    July 6, 2014 4:34 pm


    Thanks for your long reply to my comment. You are correct. I got sidetracked by the first line of the third para in the post “Using hourly data, I calculated an “actual mean” as of midnight each day.” I somehow took “actual mean” to mean the temp at midnight, rather than the end point for the time duration. I did not spend enough time to understand the whole post. College statistics was a long time ago.

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