Estimating Distance

[Мастер]

From my new place, I can see a hill in the distance (on a clear day).

From the rooftop lounge, I should be able to get a good fix on the direction. Is there any decent way to estimate the distance, without moving around too much? If not, I'll go chasing the hill, but not today.

[Lance]

Google Earth has a pretty accurate measure function.

[Мастер]

Google Earth has a pretty accurate measure function.

But I have to locate the point I'm interested in on Google Earth, which is the problem I'm trying to solve in the first place.

I did find a topographic feature in Google Earth, though, and have found a cluster of 400m mountains in the distance and generally correct direction.

The working hypothesis (which I'm trying to confirm) is that I'm in a different country than the mountain.

[Мастер]

This is what I think I am looking at.

http://www.google.com/maps?ll=1.594528, ... 4&t=p&z=13

[tubeswell]

If you know the height of the hill above MSL and the height of your location above MSL, then you can measure the hypotenuse-adjacent angle from your place to the top of the hill, and use trigonometry to calculate the length of the (adjacent side) x-axis (distance to the hill)



from: http://jccc-mpg.wikidot.com/trigonometry

[Мастер]

I'm not sure I have the ability to measure the angle that precisely, at least with the equipment I've got here.

Maybe I could place another object at some short distance in the line of sight from the eye to the hilltop, and then get the distance using similar triangles.

I can probably estimate my altitude reasonably well (to within a few metres, I think). The other issue is, the height of the hills I am seeing would depend on the correctness of my identification - if they're something else, then I don't know the altitude.

However, I'm increasingly thinking my identification is correct - I have looked at some topo maps, they're in the right direction, and I don't see anything else with much height at all between here and there.

So I think I can see into another country from my spare bedroom window - can anyone else here do that?

[Мастер]

Just ran some numbers, if my identification of the hills is correct (and I am increasingly thinking it is), then a measurement error of a tenth of a degree would change the location by about 2 km.

[tubeswell]

Then take 30 measurements of the angle in order to get a representative statistical sample that you can use to estimate the population mean and variance for a normal distribution, and use that.

If the standard deviation of your initial sample looks too wide, the take a few more such statistical samples and run an analysis of variance on the whole lot.

[Мастер]

Then take 30 measurements of the angle in order to get a representative statistical sample that you can use to estimate the population mean and variance for a normal distribution, and use that.

That only works if there is no systematic bias in the measurement.

[Lianachan]

I seem to be the only one here who'd do the whole thing just with maps.

[Мастер]

I seem to be the only one here who'd do the whole thing just with maps.

But that's the issue. The whole purpose is to identify the thing I see out my window. If I knew where it was on the map, I'd be done.

I think I've pretty much managed it by verifying (from the Google top maps) that there are no other likely candidates in the general direction.

[tubeswell]

Then take 30 measurements of the angle in order to get a representative statistical sample that you can use to estimate the population mean and variance for a normal distribution, and use that.

That only works if there is no systematic bias in the measurement.

Try taking each set of samples on different days of the week, under different atmospheric conditions, wearing different glasses and drinking different types of whiskey. And then test for goodness of fit.

[Lance]

So I think I can see into another country from my spare bedroom window - can anyone else here do that?

When I lived in Wisconsin I could see Illinois from my house. Does that count?

[Мастер]

Then take 30 measurements of the angle in order to get a representative statistical sample that you can use to estimate the population mean and variance for a normal distribution, and use that.

That only works if there is no systematic bias in the measurement.

Try taking each set of samples on different days of the week, under different atmospheric conditions, wearing different glasses and drinking different types of whiskey. And then test for goodness of fit.

Goodness of fit? So see how these various conditions affect the accuracy of the measurement? If I knew the true distance, I wouldn't bother to take the measurements in the first place.

[Мастер]

So I think I can see into another country from my spare bedroom window - can anyone else here do that?

When I lived in Wisconsin I could see Illinois from my house. Does that count?

It would certainly be similar. You do need to stop at a checkpoint when entering Illinois, but instead of a passport you need an i-Pass.

[tubeswell]

When I said goodness of fit I meant test the sample to see if the SD of the sample approaches the SD in a normal distribution.

[Мастер]

When I said goodness of fit I meant test the sample to see if the SD of the sample approaches the SD in a normal distribution.

It would approach the standard deviation of any normal distribution which has that same standard deviation. A normal distribution can have any standard deviation you choose.

[tubeswell]

I will have to consult my old study notes for stats. My recollection is that if you suspect that your sample statistics are skewed or not normally distributed, you test the sample stats for goodness of fit. However, the last time I studied all this at uni was 1989, so I might well be a bit rusty.

[Мастер]

I will have to consult my old study notes for stats. My recollection is that if you suspect that your sample statistics are skewed or not normally distributed, you test the sample stats for goodness of fit.

Ah yes, one could do that. A normal distribution has a skewness of zero. However, the thing I need to know is not whether my observations come from a skewed, kurtotic, or otherwise non-normal distribution, but whether they are unbiased - is the mean or expected value of my observation equal to the true value of the thing being measured. Testing for normality won't tell me that.

But, I am pretty sure that I have answered the question using a method similar to Lianachan's - I looked at the topographic map, searched in the general direction of these hills for anything that looked like a tall hill within a whole bunch of km, and I only found one place.

[tubeswell]

I will have to consult my old study notes for stats. My recollection is that if you suspect that your sample statistics are skewed or not normally distributed, you test the sample stats for goodness of fit.

Ah yes, one could do that. A normal distribution has a skewness of zero. However, the thing I need to know is not whether my observations come from a skewed, kurtotic, or otherwise non-normal distribution, but whether they are unbiased - is the mean or expected value of my observation equal to the true value of the thing being measured. Testing for normality won't tell me that.

But my earlier proposition to take several (systematically random) sample sets of measurements, and then analyse the variance between the samples could give more confidence about the mean and SD in the statistics, and thus by inference in the population - could it not?

(I acknowledge that all this is probably tediously academic at this point).

[Enzo]

The trig function works horizontally, but not from one window. If you could look at it from a known distance to the side, a rooftop next door perhaps, and note the difference in angle compared to your window view, you can calculate the diference in angle int a distance figure. Same parallax process we use for parsecs. The Macsec. A Mactep-ocentric parallax of some amount.

[Мастер]

But my earlier proposition to take several (systematically random) sample sets of measurements, and then analyse the variance between the samples could give more confidence about the mean and SD in the statistics, and thus by inference in the population - could it not?

Assuming that each observation is an independent draw (probably reasonable, as long as I'm not changing the process or any equipment used isn't slowly wearing out), then I could estimate the mean of the population (which in this case would be the distribution of draws from this observational process) quite accurately with enough draws. The standard error of the mean would go down at the rate of the square root of the number of observations.

But that would then tell me the mean value of a measurement made by this process. Whether the mean of a measurement by this method is equal to the actual distance or not is another question. If the method produces unbiased results (there are errors, but the "average" error is zero), then it would. If the errors are biased (i.e., on average, they are not equal to zero), then such a procedure wouldn't tell us that.

If I could calibrate the whole procedure using a large number of observations of an object whose distance is known, then I could determine whether the measurement is biased or not. Assuming nothing changes when the equipment/whatever is moved/rotated/whatever to measure the new object.

The trig function works horizontally, but not from one window. If you could look at it from a known distance to the side, a rooftop next door perhaps, and note the difference in angle compared to your window view, you can calculate the diference in angle int a distance figure. Same parallax process we use for parsecs. The Macsec. A Mactep-ocentric parallax of some amount.

I actually thought about something like this, but not so much using trig. I know the location of my window quite accurately, and I can line up other objects in line with the mountain, whose location is also known. Draw a straight line on the map (we're not talking such big distances that the curvature of the earth should matter) from my location to the known location, and extrapolate beyond - the mountain is on that line. Do the same thing from another location.

Just have to find another suitable location - not so many tall buildings around this part of town, and those that there are, I mostly don't have access to them.

[Heid the Ba]

So I think I can see into another country from my spare bedroom window - can anyone else here do that?

I can't, but I can see another county.

In his Aachen days the good doctor may have been able to, he had two other countries within a few kilometers.

[MM_Dandy]

Can't even see Iowa from my house - even though it's ~1 mile away. Too many other houses and trees in the way. I can't speak as to your local conditions, Мастер, but here I often vastly underestimate distances to landmarks going on visual observation alone. Once you know where something is, of course, you know how far it is despite how it looks.

Anecdote time!

Once, as a teenager, we (my family and I) spotted smoke from a distant fire almost due east from my parent's farm. I remember thinking that it wasn't far away. I figured maybe 4 miles. We decided to drive over and take a look. But after driving the first four miles, the smoke appeared only marginally closer. Not only was it much further away than I thought, but it turned out to be a lot harder to pin point the source of the fire than we had anticipated. When we finally tracked it down, it turned out to be about 10 miles away and a good deal further north!

[Lianachan]

But, I am pretty sure that I have answered the question using a method similar to Lianachan's - I looked at the topographic map, searched in the general direction of these hills for anything that looked like a tall hill within a whole bunch of km, and I only found one place.

Yeay! Although when I said "just with maps", I took it as a given that a compass would also be involved. It is when I'm trying to pinpoint stuff when distance/hill walking, finding archaeological sites, etc. I have modern, fangled GPS contraptions, but for safety reasons I like to maintain the ability to do it Old Skool.