GT2014/15 - Don't Follow the Lights

Interesting concept, thanks for sharing

I guess you would eventually start measuring around the edges of each individual rock - and the exact waterline would become a problem (aside from issues arising from the tide and waves).

It reminds me of a meme I once saw:


Obviously an incorrect use of limits as the sides of a circle are not defined as infinitely closer squares which is where the difference between the 3.14 and 4 comes in. But still - similar concept.

At the risk of getting far too technical after a few glasses of wine:

The coastline paradox would only apply if your means of measurement also became increasingly accurate as you increased the sampling frequency.

In the case of Ghaz and his GPS in the Berg, the accuracy of his GPS track in approximating the "true" distance he has walked is limited by his GPS's ability to accurately determine his exact position.

If, for example, the GPS was at 50m accuracy, sampling once every minute and Ghaz was walking a dead-straight line. At his starting point, there is equal probability that his GPS will mark him anywhere with a 50m radius of his actual position (including 50m ahead). Unless he covers at least 100m before the next sampling point 1 min from now (6km/h) there is a non-zero probability that his GPS will mark his next position as behind his measured starting point. Repeat this over the entire day and you'll have many GPS tracks doubling back on themselves.

So, depending on the GPS accuracy and the "bendiness" in the actual route walked, I propose that there is an optimal speed, below which the GPS over-estimates the true distance (due to false measurement spikes) and above which the GPS under-estimates the true distance (due to linearly-approximating the bends).

The exact function for the estimation of this optimal speed based on the accuracy and bendiness parameters, I will leave to Ghaz and many years of field study and scientific observation in the mountains!

I would like to propose an experiment:

Equipment required:
1) 2 GPS devices (preferably the same brand and model)
2) A trail of around 6km, preferably away from cliffs and high buildings in order to increase GPS accuracy
3) 2 hikers

Method:
1) At the starting point, once both devices have found the location to a relatively high degree of accuracy, the existing track and trip computer must be reset
2) Both GPSs must be set to the same recording frequency
3) Walk the first 3km of the trip and stop. Compare the total distance walked according to each device, and then save the track and clear both devices
4) Now set the 1 device to maximum frequency and the other to minimum frequency.
5) Complete the trail and once again save the track
6) Use the first track as a control and evaluate the results by comparing the distances on the second track

I would be interested to see how the results come out.

For the GT, my device did generally claim accuracy of between 3 and 5m. I see the specific co-ordinates vary in distance between 14m and 2m, but are mostly around 7m. In theory, the most accurate track would be a measure of every time one takes a step, provided the GPS was sufficiently accurate to do this. On the steeper slopes I would have sometimes done close to double the direct distance on the way up the slopes. Ndumeni Dome would be a good example of this, although my GPS track most certainly doesn't show it.

Here is some empirical data to add to the distance issue - using an isentropic surface on GE near Erg Chech in Algeria to simplify the terrain and make a nice back drop.


The image below shows a zig-zag route which adds up to 17.7 km. If a GPS happened to sampled the position at every corner of the zig-zag then the distance walked and the GPS distance should be the same.



The image below has the straight line distance as well as the original (walked) zig-zag line shown. The straight line distance is 8.3 km. If the GPS only sampled the position of the hiker twice, once at the beginning and once at the end, then the distance would be under estimated by more than a factor of two.





The image below shows the original route as well as a route that comes about through sampling about every second turning point on the original route. Here the distance measured by the GPS would be 9.4 km.



For illustration sake, if the GPS were to sample once an hour and the hiker took 12 hours to do the original route, then the GPS would record something close to 17 km. If the GPS were to sample once an hour and the hiker took 1 hour to do the original route (this is a 25th century GT record contender capable of doing 17 km an hour!), then the GPS would record only 8.3 km.

If the hiker took four hours and the GPS were set to sample once an hour, then the distance would be reported as 9.4 km.

The sampling rate therefore has a drastic effect on the distance measured. The number of sampled positions (and therefore total distance) is a function of the speed of the hiker.

Stijn wrote: At the risk of getting far too technical after a few glasses of wine:

The coastline paradox would only apply if your means of measurement also became increasingly accurate as you increased the sampling frequency.

With the benefit of no wine and a cup of coffee - I would expect that the matrix of x,y errors on the GPS position on a Cartesian coordinate would sum to zero and therefore, on a sufficiently long track, not influence the bias in distance. Surely Stijn?

mnt_tiska wrote:

Stijn wrote: At the risk of getting far too technical after a few glasses of wine:

The coastline paradox would only apply if your means of measurement also became increasingly accurate as you increased the sampling frequency.

With the benefit of no wine and a cup of coffee - I would expect that the matrix of x,y errors on the GPS position on a Cartesian coordinate would sum to zero and therefore, on a sufficiently long track, not influence the bias in distance. Surely Stijn?

Hmmm... I'm not so sure. Since all the errors accumulate in absolute terms, errors in different directions don't cancel each other out.

Take the extreme case of standing still while your GPS keeps sampling. As long as the GPS accuracy is not 100%, your distance will gradually increase as it adds the distance between all the slightly inaccurate sampled points, despite the fact that your true distance is not increasing at all.

While moving, this effect has more of an over-estimation effect the less accurate your GPS readings and the slower you go.

So using your example, based on a given sampling frequency, you need to be going slowly enough to be relatively certain that you'll have measured observations at each of the corner points.

But go too slowly and the inaccuracy of your GPS will start adding cumulative errors around your true path, which will over-estimate your true distance.

Hence the optimal speed I mentioned above, dependent on GPS accuracy, sample frequency and number of turns in your path.

Here's an extreme example, from a scramble we did up the Jonkershoek Twins recently. We were in a narrow gully with 200m cliffs around us, so the GPS accuracy was particularly poor. My watch sampling rate was also particularly high (1s) and we were moving slowly, allowing many GPS errors to accumulate false distance.

Actual route taken is indicated in red. The green/yellow line is the route the GPS tracked us on. Probably over-estimating our distance by more than a km.



Granted, with the much higher accuracy Ghaz would have had on the escarpment, this issue would only have added false distance if he was standing still with the GPS on for long periods of time.

Stijn wrote: this issue would only have added false distance if he was standing still with the GPS on for long periods of time.

Ghaz - Stijn has just cracked how you can break your annual distance record effortlessly this year - head up to the escarpment with a stash of batteries for your GPS, pitch your tent and sit still!

Next Grand Challenge is to do the GT with a surveyor's wheel in less than 230 km.....