# Confused with the effect number of decimals at the output of transducer

There is an ultrasonic device which measures and outputs both wind speed and direction and has a very long manual. But to keep it short and summarize, the accuracy and the resolution for the wind direction is given in the following table: As default, the device outputs the wind direction as for instance: 355.2 which is the direction with one decimal. Now the number of decimals can be user defined and be like 355.467 ect.

My questions are about comparing the logging of the output with one decimal versus with three decimals.

• In terms of accuracy would increasing the number of decimals make any difference in the accuracy?
• And in terms of resolution would increasing the number of decimals make any difference in the resolution?
• If neither, what parameter would improve doing so?
• It tells you right there ... accuracy 1 degree. How could adding resolution below 0.1 degree improve that? Mar 25, 2020 at 11:51

In terms of accuracy would increasing the number of decimals make any difference in the accuracy?

The accuracy is specified as 1° up to wind speeds of 35 m/s so, acquire the number with enough resolution to avoid making the accuracy any worse than 1°. In other words you would need one decimal place to be confident.

And in terms of resolution would increasing the number of decimals make any difference in the resolution?

The number of decimal places is the resolution. It doesn't give more accuracy but it does allow you to see small changes of wind speed if that is important to you.

• Isn't there a thread/discussion here on precision vs accuracy vs resolution? Seems to me this topic comes up periodically, in various forms. Mar 25, 2020 at 11:43
• Doesn't it say 1 degree, not 1 percent? Mar 25, 2020 at 12:59
• @jonathanjo yes it does, my mistake. I'll fix. Mar 25, 2020 at 13:07

That selectable output format is probably just meant as convenience to the user.

• Of course, if you used less decimals than your sensor had resolution, you'd lose information in that quantization step.
• Your 0.1° resolution at accurracies worse than 1° is of course also only meaningful if you have some information on how the error between reported angle and actual angle is distributed, in which you can use that to implement an estimator for the actual angle based on multiple observations.
• To that end, it might be more useful to have more digits, even if they're just the result of noise. Or it might not, if your estimation method boils down to be mathematically equivalent to first rounding all numbers.

We might need to talk about what "resolution" is: that's easy if you have say simply a piece of plastic mounted on a rotatable platform that you hold into the wind, it rotates so that it's in the direction of the wind, and you've got some optical encoder linked to the platform that gives you one out of 360 possible angles. 1° resolution, easy. The measurement noise will be simply the quantization noise (i.e. assuming wind directions are equiprobable, you'll be off by ¼° on average, simply because of your resolution).

This device is way more complex. It's something that measures the direction of wind by estimating the doppler shift in orthogonal directions and then calculating a likely direction of wind that led to these estimates of doppler shift.

Doing that math, you can often get arbitrarily close to the real value, if you can observe the same phenomenon (wind direction) just for long enough (for details on what's possible with an unbiased estimator: see Cramer-Rao Lower Bound).

However, in a real system, you have a limited amount of observations, which limits the amount of noise due to actual physical uncertainty (wind isn't 100% constant), measurement errors (amplifier temperature, for example), digitization, doppler estimation (that's spectral estimation, there's libraries full of books on that), and from that, direction estimation. All these steps can introduce new errors!

So, if you're designing a system, you'd go through all these things and say, OK, in the end, my estimate of direction will have a variance $$\\sigma^2\$$, i.e. a mean squared deviaion from its mean, and that gives us an interval that we can be pretty sure about being in. In a normally distributed variable, more than 95% of values fall into $$\2\sigma\$$ around the mean!

So, probably, they used some statistics to say (for example) something like

if we don't promise we're more accurate than $$\2\sigma\$$, which is 0.1°, then our measurement is right 95% of time! Let's call that the resolution.

I don't know what you're measuring, but 0.1° is pretty impressive in angular resolution for anything that isn't solid.

• +1 for Cramer-Rao Lower Bound! Mar 25, 2020 at 11:49

There is nothing to be gained by increasing the resolution beyond the accuracy quoted. Taking the example you quote, the figure 355.467 is spurious; the true value lies between 354.467 and 356.467. Accordingly, it is more realistic to quote the value as 355.5, and I would prefer to quote simply 355. In my experience, wind is turbulent to the extent that it is difficult to define a direction to single-degree value, and averaging results over a period of time is complicated by random gusts.

I've used a similar kind of device from a different manufacturer, and have done a fair amount of weather data logging for various clients. It's hard to believe there's much meaning even in 0.1° measurements of direction for a single sensor in a fluid. Is your device oriented to within 0.1°?

My understanding of the functioning of this kind of device is that it works out the speed of (ultra)sound between three outputs and three inputs by doppler, then works out the direction of the wind which would give that effect. As such, the outputs are the results of pretty tricky calculations. When they say it has a resolution of 0.1°, what does it mean? I can only assume it means you put it in a wind tunnel with laminal flow and if you rotate the device by 0.1°, you get a change of output of 0.1°.You could print as many decimals as you cared for -- and you would, while designing the machine -- but what does it mean for weather?

A quick reading of the manual suggests you might need to correct wind speeds according to "acoustic virtual temperature". Are you doing this?

Depending on application, I would decid according to one of two principles:

• Log what I have If the device claims 0.1, log 0.1. You can then worry about it later.
• Log what I can replace If most anemometers give whole degrees, log whole degrees. Then I can't have any dependencies on this particular model.

If it was me I'd just log whole degrees.

• Isn't that the "acoustic virtual temperature" is a function of wind speed? I couldn't see/find the wind speed dependence on "acoustic virtual temperature". Where does it show such relation or correction? Thanks Mar 25, 2020 at 13:15
• "The deviation of the measured "acoustic temperature" from the real air temperature shows linear dependence on the absolute humidity level of the air." linked manual p12. "With a 6 K temperature deviation there is thus a measuring error of approx. 1 % for the wind velocity " linked manual p19. Mar 25, 2020 at 13:53
• I see but if I want to correct it I couldn't find a formula where it shows the wind speed is a function of virtual temperature or humidity is a function of virtual temperature. And it doesnt say whether it already corrects it via its microprocessor. Thanks I will look into these, Mar 25, 2020 at 14:03
• p19 I believe they are saying there is a thermal expansion of 0.17 %/K, ie 170 ppm, but that seems pretty large en.wikipedia.org/wiki/Thermal_expansion Mar 25, 2020 at 14:42