For any coefficient matrix A(t), the equation

y’ = A*y + f’ – A*f

has y=f as a solution. A perfect emulator. But as I showed above, the error propagation is given by the homogeneous part y’ = A*y. And that could be anything at all, depending on choice of A. Sharing a common solution does not mean that two equations share error propagation. So it’s not OK.

A post elsewhere that highlights the problem Nick is trying to address

“how is it that we can reasonably accurate calculate GMST with only about 60 gauges? I know that ATTP has had at least one blog post in that regard. Now, I think that error improves as the (inverse) square root of the number of gauges. The average is twice as accurate for N = 3,600, not proportional to the square root of N but proportional to the inverse square root of N.”

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GMST is such a fraught concept.

Problem one is the definition of the surface on a mixed changing atmospheric world (variable water vapour) plus a mixed solid/liquid “surface of variable height and depth on top of an uneven shape with long term variability in the spin and torque and inclination of the world plus the variation in distance from the heating element plus variation in the shade from the satellite at times and albedo variation from clouds and volcanic emissions and ice and dust storms and heating from volcanic eruptions and CO2 emmision and human CO2 emissions.

Phew.

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We could get around this partly by measuring solar output, albedo change and earth output from space by satellites and just using a planetary emmision temperature as a substitute for GMST.

You could actually compute what the temperature should be at any location on earth purely by it’s elevation, time of year and orientatation in space to the sun without using a thermometer.

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In a model world, barring inbuilt bias, one only ever needs one model thermometer. There can be no error. Using 3600 does not improve the accuracy.

In a model world allowing a standard deviation for error will lead to a possible Pat Frank scenario. The dice can randomly throw +4W/m-2 for ever. Having thrown one head is no guarantee that the next throw or the next billion throws will not be a head.

Using 3600 instead of 60 does not improve the accuracy at all. It improves the expectation of where the accuracy should be is all. While they look identical accuracy and expectation of accuracy are two completely different things. Your statement on probability is correct.

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Finally this presupposes a model world and temperature and reasonable behaviour. Thermometers break,or degrade over time, people enter results wrongly,or make them up or take them at the wrong time of day or average them when missing ( historical). The accuracy changes over time. They only cover where people can get to easily, like looking for your keys under the streetlight, spacial, height, sea, polar, desert, Antarctica etc. Collating the information in a timely manner, not 3 months later when it all comes in. Are 3600 thermometers in USA better than 60 scattered around the world.

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60 is a good number adequately sited for an estimation. 3600 is a lot better. As Paul said any improvement helps modelling tremendously.

Not having a go at you, just pointing out the fraughtness