Guest Essay by Kip Hansen
Introduction:
Temperature and Water Level (MSL) are two hot topic measurements being widely bandied about and vast sums of money are being invested in research to determine whether, on a global scale, these physical quantities — Global Average Temperature and Global Mean Sea Level — are changing, and if changing, at what magnitude and at what rate. The Global Averages of these ever-changing, continuous variables are being said to be calculated to extremely precise levels — hundredths of a degree for temperature and millimeters for Global Sea Level — and minute changes on those scales are claimed to be significant and important.
In my recent essays on Tide Gauges, the question of the durability of original measurement uncertainty raised its toothy head in the comments section.
Here is the question I will try to resolve in this essay:
If original measurements are made to an accuracy of +/- X (some value in some units), does the uncertainty of the original measurement devolve on any and all averages – to the mean – of these measurements?
Does taking more measurements to that same degree of accuracy allow one to create more accurate averages or “means”?
My stated position in the essay read as follows:
If each measurement is only accurate to ± 2 cm, then the monthly mean cannot be MORE accurate than that — it must carry the same range of error/uncertainty as the original measurements from which it is made. Averaging does not increase accuracy.
It would be an understatement to say that there was a lot of disagreement from some statisticians and those with classical statistics training.
I will not touch on the subject of precision or the precision of means. There is a good discussion of the subject on the Wiki page: Accuracy and precision .
The subject of concern here is plain vanilla accuracy: “accuracy of a measurement is the degree of closeness of measurement of a quantity to that quantity’s true value.” [ True value means is the actual real world value — not some cognitive construct of it.)
The general statistician’s viewpoint is summarized in this comment:
“The suggestion that the accuracy of the mean sea level at a location is not improved by taking many readings over an extended period is risible, and betrays a fundamental lack of understanding of physical science.”
I will admit that at one time, fresh from university, I agreed with the StatsFolk. That is, until I asked a famous statistician this question and was promptly and thoroughly drummed into submission with a series of homework assignments designed to prove to myself that the idea is incorrect in many cases.
First Example:
Let’s start with a simple example about temperatures. Temperatures, in the USA, are reported and recorded in whole degrees Fahrenheit. (Don’t ask why we don’t use the scientific standard. I don’t know). These whole Fahrenheit degree records are then machine converted into Celsius (centigrade) degrees to one decimal place, such as 15.6 °C.
This means that each and every temperature between, for example, 72.5 and 71.5 °F is recorded as 72 °F. (In practice, one or the other of the precisely .5 readings is excluded and the other rounded up or down). Thus an official report for the temperature at the Battery, NY at 12 noon of “72 °F” means, in the real world, that the temperature, by measurement, was found to lie in the range of 71.5 °F and 72.5 °F — in other words, the recorded figure represents a range 1 degree F wide.
In scientific literature, we might see this in the notation: 72 +/- 0.5 °F. This then is often misunderstood to be some sort of “confidence interval”, “error bar”, or standard deviation.
It is none of those things in this specific example of temperature measurements. It is simply a form of shorthand for the actual measurement procedure which is to represent each 1 degree range of temperature as a single integer — when the real world meaning is “some temperature in the range of 0.5 degrees above or below the integer reported”.
Any difference of the actual temperature, above or below the reported integer is not an error. These deviations are not “random errors” and are not “normally distributed”.
Repeating for emphasis: The integer reported for the temperature at some place/time is shorthand for a degree-wide range of actual temperatures, which though measured to be different, are reported with the same integer. Visually:
Even though the practice is to record only whole integer temperatures, in the real world, temperatures do not change in one-degree steps — 72, 73, 74, 72, 71, etc. Temperature is a continuous variable. Not only is temperature a continuous variable, it is a constantly changing variable. When temperature is measured at 11:00 and at 11:01, one is measuring two different quantities; the measurements are independent of one another. Further, any and all values in the range shown above are equally likely — Nature does not “prefer” temperatures closer to the whole degree integer value.
[ Note: In the U.S., whole degree Fahrenheit values are converted to Celsius values rounded to one decimal place –72°F is converted and also recorded as 22.2°C. Nature does not prefer temperatures closer to tenths of a degree Celsius either. ]
While the current practice is to report an integer to represent the range from integer-plus-half-a-degree to integer-minus-half-a-degree, this practice could have been some other notation just as well. It might have been just report the integer to represent all temperatures from the integer to the next integer, as in 71 to mean “any temperature from 71 to 72” — the current system of using the midpoint integer is better because the integer reported is centered in the range it represents — this practice, however, is easily misunderstood when notated 72 +/- 0.5.
Because temperature is a continuous variable, deviations from the whole integer are not even “deviations” — they are just the portion of the temperature measured in degrees Fahrenheit normally represented by the decimal fraction that would follow the whole degree notation — the “.4999” part of 72.4999°F. These decimal portions are not errors, they are the unreported, unrecorded part of the measurement and because temperature is a continuous variable, must be considered evenly spread across the entire scale — in other words, they are not, not, not “normally distributed random errors”. They only reason they are uncertain is that even when measured, they have not been recorded.
So what happens when we now find the mean of these records, which, remember, are short-hand notations of temperature ranges?
Let’s do a basic, grade-school level experiment to find out…
We will find the mean of a whole three temperatures; we will use these recorded temperatures from my living room:
11:00 71 degrees F 12:00 72 degrees F 13:00 73 degrees F
As discussed above, each of these recorded temperatures really represent any of the infinitely variable intervening temperatures, however I will make this little boxy chart:
Here we see each hour’s temperature represented as the highest value in the range, the midpoint value of the range (the reported integer), and as the lowest value of the range. [ Note: Between each box in a column, we must remember that there are an infinite number of fractional values, we just are not showing them at this time. ] These are then averaged — the mean calculated — left to right: the three hour’s highest values give a mean of 72.5, the midpoint values give a mean of 72, and the lowest values give a mean of 71.5.
The resultant mean could be written in this form: 72 +/- 0.5 which would be a short-hand notation representing the range from 71.5 to 72.5.
The accuracy of the mean, represented in notation as +/- 0.5, is identical to the original measurement accuracy — they both represent a range of possible values.
Note: This uncertainty stems not from the actual instrumental accuracy of the original measurement, which is a different issue and must be considered additive to the accuracy discussed here which arises solely from the fact that measured temperatures are recorded as one-degree ranges with the fractional information discarded and lost forever, leaving us with the uncertainty — a lack of knowledge — of what the actual measurement itself was.
Of course, the 11:00 actual temperature might have been 71.5, the 12:00 actual temperature 72, and the 13:00 temperature 72.5. Or it may have been 70.5, 72, 73.5.
Finding the means kiddy-corner gives us 72 for each corner to corner, and across the midpoints still gives 72.
Any combination of high, mid-, and low, one from each hour, gives a mean that falls between 72.5 and 71.5 — within the range of uncertainty for the mean.
Even for these simplified grids, there are many possible combinations of one value from each column. The means of any of these combinations falls between the values of 72.5 and 71.5.
There are literally an infinite number of potential values between 72.5 and 71.5 (someone correct me if I am wrong, infinity is a tricky subject) as temperature is a continuous variable. All possible values for each hourly temperature are just as likely to occur — thus all possible values, and all possible combinations of one value for each hour, must be considered. Taking any one possible value from each hourly reading column and finding the mean of the three gives the same result — all means have a value between 72.5 and 71.5, which represents a range of the same magnitude as the original measurement’s, a range one degree Fahrenheit wide.
The accuracy of the mean is exactly the same as the accuracy for the original measurement — they are both a 1-degree wide range. It has not been reduced one bit through the averaging process. It cannot be.
Note: For those who prefer a more technical treatment of this topic should read Clyde Spencer’s “The Meaning and Utility of Averages as it Applies to Climate” and my series “The Laws of Averages”.
And Tide Gauge Data?
It is clear that the original measurement accuracy’s uncertainty in the temperature record arises from the procedure of reporting only whole degrees F or degrees C to one decimal place, thus giving us not measurements with a single value, but ranges in their places.
But what about tide gauge data? Isn’t it a single reported value to millimetric precision, thus different from the above example?
The short answer is NO, but I don’t suppose anyone will let me get away with that.
What are the data collected by Tide Gauges in the United States (and similarly in most other developed nations)?
The Estimated Accuracy is shown as +/- 0.02 m (2 cm) for individual measurements and claimed to be +/- 0.005 m (5 mm) for monthly means. When we look at a data record for the Battery, NY tide gauge we see something like this:
| Date Time | Water Level | Sigma |
| 9/8/2017 0:00 | 4.639 | 0.092 |
| 9/8/2017 0:06 | 4.744 | 0.085 |
| 9/8/2017 0:12 | 4.833 | 0.082 |
| 9/8/2017 0:18 | 4.905 | 0.082 |
| 9/8/2017 0:24 | 4.977 | 0.18 |
| 9/8/2017 0:30 | 5.039 | 0.121 |
Notice that, as the spec sheet says, we have a record every six minutes (1/10th hr), water level is reported in meters to the millimeter level (4.639 m) and the “sigma” is given. The six-minute figure is calculated as follows:
“181 one-second water level samples centered on each tenth of an hour are averaged, a three standard deviation outlier rejection test applied, the mean and standard deviation are recalculated and reported along with the number of outliers. (3 minute water level average)”
Just to be sure we would understand this procedure, I emailed CO-OPS support [ @ co-ops.userservices@noaa.gov ]:
To clarify what they mean by accuracy, I asked:
When we say spec’d to the accuracy of +/- 2 cm we specifically mean that each measurement is believed to match the actual instantaneous water level outside the stilling well to be within that +/- 2 cm range.
And received the answer:
That is correct, the accuracy of each 6-minute data value is +/- 0.02m (2cm) of the water level value at that time.
[ Note: In a separate email, it was clarified that “Sigma is the standard deviation, essential the statistical variance, between these (181 1-second) samples.” ]
The question and answer verify that both the individual 1-second measurements and the 6-minute data value represents a range of water level 4 cm wide, 2 cm plus or minus of the value recorded.
This seemingly vague accuracy — each measurement actually a range 4 cm or 1 ½ inches wide — is the result of the mechanical procedure of the measurement apparatus, despite its resolution of 1 millimeter. How so?
NOAA’s illustration of the modern Acoustic water level tide gauge at the Battery, NY shows why this is so. The blow-up circle to the top-left shows clearly what happens at the one second interval of measurement: The instantaneous water level inside the stilling well is different than the instantaneous water level outside the stilling well.
This one-second reading, which is stored in the “primary data collection platform” and later used as part of the 181 readings averaged to get the 6-minute recorded value, will be different from the actual water level outside the stilling well, as illustrated. Sometimes it will be lower than the actual water level, sometimes it will be higher. The apparatus as a whole is designed to limit this difference, in most cases, at the one second time scale, to a range of 2 cm above or below the level inside the stilling well — although some readings will be far outside this range, and will be discarded as “outliers” (the rule is to discard all 3-sigma outliers — of the set of 181 readings — from the set before calculating the mean which is reported as the six-minute record).
We cannot regard each individual measurement as measuring the water level outside the stilling well — they measure the water level inside the stilling well. These inside-the-well measurements are both very accurate and precise — to 1 millimeter. However, each 1-second record is a mechanical approximation of the water level outside the well — the actual water level of the harbor, which is a constantly changing continuous variable — specified to the accuracy range of +/- 2 centimeters. The recorded measurements represent ranges of values. These measurements do not have “errors” (random or otherwise) when they are different than the actual harbor water level. The water level in the harbor or river or bay itself was never actually measured.
The data recorded as “water level” is a derived value – it is not a direct measurement at all. The tide gauge, as a measurement instrument, has been designed so that it will report measurements inside the well that will be reliably within 2 cm, plus or minus, of the actual instantaneous water level outside the well – which is the thing we wish to measure. After taking 181 measurements inside the well, throwing out any data that seems too far off, the remainder of the 181 are averaged and reported as the six-minute recorded value, with the correct accuracy notation of +/- 2 cm — the same accuracy notation as for the individual 1-second measurements.
The recorded value denotes a value range – which must always be properly noted with each value — in the case of water levels from NOAA tide gauges, +/- 2 cm.
NOAA quite correctly makes no claim that the six-second records, which are the means of 181 1-second records, have any greater accuracy than the original individual measurements.
Why then do they make a claim that monthly means are then accurate to +/- 0.005 meters (5 mm)? In those calculations, the original measurement accuracy is simply ignored altogether, and only the reported/recorded six-minute mean values are considered (confirmed by the author) — the same error that is made as with almost all other large data set calculations, applying the inapplicable Law of Large Numbers.
Accuracy, however, as demonstrated here, is determined by the accuracy of the original measurements when measuring a non-static, ever-changing, continuously variable quantity and which is then recorded as a range of possible values — the range of accuracy specified for the measurement system — and cannot be improved when (or by) calculating means.
Take Home Messages:
- When numerical values are ranges, rather than true discrete values, the width of the range of the original value (measurement in our cases) determines the width of the range of any subsequent mean or average of these numerical values.
- Temperatures calculated from ASOS stations however are recorded and reported temperatures as ranges 1°F wide (0.55°C), and such temperatures are correctly recorded as “Integer +/- 0.5°F”. The means of these recorded temperatures cannot be more accurate than the original measurements –because the original measurement records themselves are ranges, the means must be denoted with the same +/- 0.5°F.
- The same is true of Tide Gauge data as currently collected and recorded. The primary record of 6-minute-values, though recorded to millimetric precision, are also ranges with an original accuracy of +/- 2 centimeters. This is the result of the measurement instrument design and specification, which is that of a sort-of mechanical averaging system. The means of tide gauge recorded values cannot be made more accurate the +/- 2 cm — which is far more accurate than needed for measuring tides and determining safe water levels for ships and boats.
- When original measurements are ranges, their means are also ranges of the same magnitude. This fact must not be ignored or discounted; doing so creates a false sense of the accuracy of our numerical knowledge. Often the mathematical precision of a calculated mean overshadows its real world, far fuzzier accuracy, leading to incorrect significance being given to changes of very small magnitude in those over-confident means.
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Author’s Comment Policy:
Thanks for reading — I know that this will be a difficult concept for some. For those, I advise working through the example themselves. Use as many measurements as you have patience for. Work out all the possible means of all the possible values of the measurements, within the ranges of those original measurements, then report the range of the means found.
I’d be glad to answer your questions on the subject, as long as they are civil and constructive.
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Thank you for all the hard work.
I worked with IMD in Pune/India [prepared formats to transfer data on to punched cards as there was no computer to transfer the data directly]. There are two factors that affect the accuracy of data, namely:
Prior to 1957 the unit of measurement was rainfall in inches and temperature in oF and from 1957 they are in mm and oC. Now, all these were converted in to mm and oC for global comparison.
The second is correcting to first place of decimal while averaging: 34.15 is 34.1; 34.16 is 34.2; 34.14 is 34.1 and 34.25 is 34.3; 34.26 is 34.3; 34.24 is 34.2
Observational error: Error in inches is higher than mm and Error in oC is higher than oF
These are common to all nations defined by WMO
Dr. S. Jeevananda Reddy
Thank you for this information. I have always suspected the reported accuracy of many averaged numbers were simply impossible. This helps to clarify my suspicions. I also do not understand how using 100 year old measurements mixed with modern ones can result in the high accuracy stated in many posts. They seem to just assume that a lot of values increases the final accuracy regardless of the origin and magnitude of the underlying uncertainties.
Kip says: “If each measurement is only accurate to ± 2 cm, then the monthly mean cannot be MORE accurate than that — it must carry the same range of error/uncertainty as the original measurements from which it is made. Averaging does not increase accuracy.”
…
WRONG!
…
the +/- 2cm is the standard deviation of the measurement. This value is “sigma of x ” in the equation for the standard error of the estimator of the mean:
https://www.bing.com/images/search?view=detailV2&ccid=CYUOXtuv&id=B531D5E2BA00E15F611F3DAEC1B85110014F74C6&thid=OIP.CYUOXtuvcFogpL3jEnQw_gEsBg&q=standard+error&simid=608028072239301597&selectedIndex=1
…
The error bars for the mean estimator depends on the sqrt of “N”
It may be worth remembering – no calculated figure is entitled to more significant figures (accuracy) than the data used in the calculation.
Kip is correct if the temperature never deviates from 72degF +/- 0.5degF. You will just write down 72 degF and the error will indeed be has he indicates.
Fortunately the temperature varies far more than that. One day, the temperature high/ow is 72/45 from 71.5 true and 45.6 true, the next day it is 73/43 from 72.3 true and 44.8 true, the next day it is 79/48 from 79.4 true and 47.9 true, and so on. The noise that is the difference between the true and recorded measurement has an even distribution as he notes, but can be averaged as long as the underlying signal swings bigger than the resolution of 1degF.
The Central Limit is a real thing. You average together a bunch of data with rectangular distribution you get a normal distribution. Go ahead and look at the distribution of a 6 sided dice. With one dice it’s rectangular. With two dice it’s a triangle. Add more and more dice and it’s a normal distribution.
Fortunately the signal varies by more than the 1 bit comparator window for the sigma-delta A/D and D/A converters in your audio and video systems, which operate on similar principles. It would be quite obvious to your ears if they failed to work. (yes, they do some fancy feedback stuff to make it better, but you can get a poor man’s version by simple averaging. I’ve actually designed and built the circuits and software to do so)
Peter
I’m also somewhat surprised that they do not use ‘banker’s rounding’ (google it). Not using BR adds an upwards bias with a large amount of data, which is why banks do use it.
“In scientific literature, we might see this in the notation: 72 +/- 0.5 °F. This then is often misunderstood to be some sort of “confidence interval”, “error bar”, or standard deviation”
The confusion is understandable? It’s been sixty years, but I’m quite sure they taught me at UCLA in 1960 or so that the 72 +/- notation is used for both precision based estimates and for cases where the real error limits are somehow known. It’s up to the reader to determine which from context or a priori knowledge?
I’d go of and research that, but by the time I got an answer — if I got an answer — this thread would be long since dead. Beside which, I’d rather spend my “How things work time” this week trying to understand FFTs.
Anyway — thanks as usual for publishing these thought provoking essays.
Kip,
You do have over a century of scientific understanding against you. And you give almost no quantitative argument. And you are just wrong. Simple experiments disprove it.
In the spirit of rounding, I took a century of Melbourne daily maxima (to 2012, a file I have on hand). They are given to 0.1°C. That might be optimistic, but it doesn’t matter for the demo. For each month, I calculated the average of the days. Then I rounded each daily max to the nearest °C, and again calculated the average. Here are the results:
As you’ll seen despite the loss of accuracy in rounding (To 0 dp), the averages of those 100 years, about 3000 days, does not have an error of order 1. In fact, the theoretical error is about 0.28/sqrt(3000)= 0.0054°C, and the sd of the differences shown is indeed 0.0062. 0.28 is the approx sd of the unit uniform distribution.
Is temperature truly infinite in the continuum like time is, or does it have quanta associated with it like radiation?
I found a sample document to read, but I haven’t extensively studied quantum mechanics yet :
TEMPERATURE IN QUANTUM DYNAMICS
ALESSANDRO
SERGI
ABSTRACT
. What is the meaning of the thermodynamical temperature in quantum mechanics? What is its role in the classical limit? What can we say about the interplay between quantum and thermal fluctuations? Can we impose a constant-temperature constraint within dynamical simulations on quantum systems as we do in simulations of classical systems?
https://www.scribd.com/mobile/document/40884849/Temperature-in-quantum-mechanics
“[ Note: In a separate email, it was clarified that “Sigma is the standard deviation, essential the statistical variance, between these (181 1-second) samples.” ]”
My guess is this isn’t as odd as it sounds. But. “essential” was probably intended to be “essentially”?. And the standard deviation is the square root of the variance? My guess is that “variance” really should have been the less rigorous term “variation” Note that the statistical property “variance” has units of the variable under discussion squared and is often a disturbingly large number compared to the actual size of the errors.
I don’t want to be pedantic, but this is a subject that I taught to laboratory technicians and engineers for many years.
Sorry MSJ, Kip is correct and you are wrong. A measurement uncertainty specification states the size of an interval around the indicated value within which the true value is thought to lie. Properly stated the MU specified should indicate the distribution type used to determine it – i.e. normal, rectangular or triangular and a confidence level – see the ISO Guide to the Expression of Uncertainty in Measurement (GUM). It is typically 2 times the standard uncertainty derived from calibration comparisons the a primary reference along with evaluation of additional sources of MU. There are always more than one.
When multiple measurements of something are made with a precise enough instrument they will invariably differ by some amount. The differences are considered random error and this can be reduced by averaging a number of measurements. But the random error is a source of uncertainty that is in addition to the instrument MU.
So, it I take 100 measurements with a +/- 2 cm instrument and get an average of 50 cm with a standard deviation of 1 cm the overall MU is +/- 2.32 cm at a 95% confidence level. [Note: there is math involved in this calculation: MU = (sqrt((2/sqrt(3))^2 + (1/sqrt(n))^2))*2].
In short, no matter how many measurements you make, the MU of the average is always greater than the MU of your instrument
Kip — I think you/we are sort of skating on thin ice over a sea of statistical sampling theory here. Sampling theory is a serious, and very complex field that is very important to industry. It’s widely used to determine things like how many samples from a production run need to be tested to have a certain degree of confidence that the run meets predefined quality standards.
I know just enough about sampling theory to know that It’s really important and mostly beyond my abilities.
In your/our case, the issue is how to sample temperature/sea level so as to get a useful/meaningful estimate of values and changes in values.
“We cannot regard each individual measurement as measuring the water level outside the stilling well”
Yes and no. Conceptually, the stilling well is just a mechanical low pass filter that removes high frequency noise caused by waves, wakes, etc. Hopefully, water level measurements made within the stilling well will yield the same value as would measurements made outside the well. With a LOT less measurement and computation.
Kip is right. Simple thought experiment. Suppose you took any large number of daily readings of an actual daily temperature of 70.4999 degrees, every reading the same, due to the constant theoretical climate. Each would be reported as 70 degrees, and any number of days averaged would result in an average of 70, when the real average is actually almost half a degree distant.
I have to agree with Nick here. When you round to nearest integer you will adjust the measurements up or down +/- 0.4999 depending on what side of the zero point they were. Over multiple measurements each individual will fall “randomly” within the +/- 0.5 intervall.
Question: How much does the worlds river water run-off, and the soils it carries, rise the oceans level
over a year’s time.
Does the increase in soil deposits around the measurement site, effect the readings. The deposits have to increase the water level over time
Nice Cutting of a Gordian knot John. Isnt it fascinating how counter intuitive statistics can sometimes be giving rise to the rather intemperent divergences of opinion seen on this thread.
For the record I am with Kip, as physicist recalling ancient lab work
When you start making claims on ‘world wide ‘ you really must face the reality of not just accuracy but ‘range ‘ Its is simply not enough to throw computing power , through models , at the issue. If you need a thousand or ten thousand measurement locations to deal with the scale, then to ‘know’ rather they ‘guess’ then that is what you should have .
And when it comes to weather or climate on a world wide scale we seem to be not even close to the coverage needed to make these measurement in a manner that supports the scientific value they are often attributed with.
One of the key statistical slight of hand tricks here is to pretend that thousands of measurement of SST are repeated measurements of the same thing, like it was some lab experiment.
In the context of AGW and attribution we are interested in assessing and attributing effects of changes in radiation flows on the surface temperature, ie for lack of better metrics we are using the ocean mixed layer as a calorimeter the measure heat energy. If we have sufficient data we can try the same thing on all water down to a certain depth and arguably get a more meaningful result.
So SST is not tens of thousand of measurements of some fixed quantity the “global temperature” since no such thing exists. There are many temperatures at different places for very good physical reasons. The “global mean temperature” is just that, it is a mean value : a statistic, it is not some physical quantity which we are taking thousands of independent measures of. In fact it is the sum which we are interested in, not the mean. The statistical confidence levels for the mean indicate how well the mean represents the sample and the confidence we can have that any given individual measurement will lie within the one or two std devs of the mean. This is not our confidence in the sum.
What we have is global array of little boxes of sea water for which we have a temperature, from which we want to estimate heat energy content. Now if we want the total energy content we will have several thousand individual measurements each with its own measurement error that we add together the get the global heat content.
There are many contributions to the uncertainty of such measurements some will average out others many be independent and thus considered orthogonal but will not average out others will be systematic and will not reduce at all.
Then we have to add changes over time of the measuring system, which itself is largely unknowable at this stage.
There is no simplistic answer like Kip’s +/-0.5 nor the usual +/- 2 s.d. which also ignores the nature of much of the uncertainty.
Re the “stilling” chamber. As Don K says there is a “low pass filter” which hopefully eliminates the effect of high frequency noise from waves and wakes. Note that depending on the location, and the design, it is unlikely to eliminate the low frequency noise from swell – easily 9 second period and in a bad location more. Further, because of the friction in the filter, the level inside will be different from that outside – when the tide is rising the interior level will be lower, and when the tide is falling the level will be higher.
As a real – ie geographical – low pass filter, I refer you to the Rip at Port Phillip Heads. Here a narrow channel connects the open ocean and Port Phillip. Consider starting from an instant when the levels inside and outside are the same. Then as the tide outside rises, water pours in through the Rip, and the tide rises inside Port Phillip. When high tide is reached outside, the water is still pouring in, and continues to do so until the outer water level has fallen to the level inside. At this time, high tide is reached inside Port Phillip, while the outside water level is approaching half tide.As tide level outside continues to fall, water commences to pour out, and the tide level inside Port Phillip starts falling. Given the range of tide outside, and the tidal range inside, the six hourly period (approximately) between high and low tide outside is offset by nearly three hours (approximately) inside.
This is on a far greater scale than that in the tide gauge, nevertheless, a similar situation. Hopefully the difference between inside and outside is rather lower!
Re the accuracy of readings. As a cadet on a cadet ship in the 1950s we had to take sea water temperatures using the bucket method. We had to read – estimate – the temperature to the nearest 0.1 degree. I cannot recall if it were in Fahrenheit or Celsius – as the thermometers were supplied by the UK Met Office I suspect that Celsius was used – I remember that in the coded message we had to insert a C or F for the readings of all temperatures. Note exactly on a degree line is easy to observe, and exactly half way between is also easy. A third or a quarter is also easy to estimate, and this gives 0.3 or 0.7, and it is fairly easy to see if it is a little more than a quarter – hence also 0.3 or 0.7, or a little less, hence 0.2 or 0.8, and if it is a tad more or less than the actual mark, then it is either 0.1 or 0.9. In good weather, the reading would be good. With a strong breeze blowing, and rain or sleet, it would be difficult to get better than to the nearest 1 degree. And in a howling gale, often it would not be possible to get the bucket in the water – it would be blown sideways so much it never reached the water.
Presume the Met Office found our reports valuable – there weren’t too many ships which reported and did so regularly every 6 hours, at 0000, 0600, 1200 and 1800 GMT.
Cook-ery with numbers
97% (+/- 98.8%) with very high confidence (because ‘Climate Science™’ is all about confidence not verification or proofs).
Your position does not seem to agree with the international guideline: Evaluation of measurement data — Guide to the expression of uncertainty in measurement
See section:
“4.4.5 For the case illustrated in Figure 2 a), it is assumed that little information is available about the input quantity t and that all one can do is suppose that t is described by a symmetric, rectangular a priori probability distribution …”
and
“4.2.3 The best estimate of σ 2(q) = σ 2 n, the variance of the mean, is given by …”
It seems as if the origin of the disagreement is related to:
“G.2.1 If Y=cX +c X +…+c X =ΣN cX and all the X are characterized by normaldistributions,…
then the resulting convolved distribution of Y will also be normal. However, even if the distributions of the Xi are not normal, the distribution of Y may often be approximated by a normal distribution because of the Central Limit Theorem. …”
It would be interesting if you, with reference to this guideline, could identify exactly the source of the disagreement. After all, if there is an error in this standard, you need to identify that error quite precisesly:
“The following seven organizations supported the development of the Guide to expression of uncertainty, which is published in their name:
BIPM: Bureau International des Poids et Measures
IEC: International Electrotechnical Commission
IFCC: International Federation of Clinical Chemistry
ISO: International Organization for Standardization
IUPAC: International Union of Pure and Applied Chemistry
IUPAP: International Union of Pure and Applied Physics
OlML: International Organization of Legal Metrology ..”
While the accuracy diacussion is interesting, it is not important to the issue of WARMING, which is the determination of rate of change. The confidence in the slope of the temperature line is certainly related to the precision, not the accuracy, and it is here where sampling makes a difference.
Enthalpy is a measurement of energy in a thermodynamic system. It is the thermodynamic quantity equivalent to the total heat content of a system. It is equal to the internal energy of the system plus the product of pressure and volume.
https://en.wikipedia.org/wiki/Enthalpy
It is the change in total energy of the earth (system) that is important. Not including atmospheric pressure and water content (humidity) in calculations is another source of error.