Thursday, July 1, 2010

GPS Maps 103: Coordinate Commandoes - Lewis and Clark

[This post is part three in a multi-part series about understanding the GPS system and its coordinate grid.]

By now you realize that there are many ways to represent geographical data, even for the same location; and most likely you have found a favorite method based upon what works for you.

Decimal Degrees: 36.9956433, -112.0059717
Decimal Minutes: N36° 59.7386, W112° 0.3583
Deg-Min-Sec: N36° 59' 44.316", W112° 0' 21.498"
And what works for you pretty much depends on how you go about navigating in the field and the ratio of paper to electronics you have in your life. The more paper – the more you fumble with degrees, minutes and seconds.

Also, after reading the previous posts, you should be able to recognize that every one of these coordinates point to the identical place – The Wave in southern Utah and Arizona. But, I can tell you, I had a mighty easier time obtaining these numbers than those intrepid souls who attempted it a hundred, fifty or even thirty years ago!

Before Electronics

Please don't think for a minute that I'm implying that Jefferson, Lewis and Clark or others I'm going to reference were the "fathers of" or even "inventors" of anything relating to navigation. They weren't even close – they were USERS. Their forte' was adventure, all-be-it with a dramatically scientific bent, but explorers none-the-less.

Calculating and quantifying position on the earth was in place as early as the ancient Babylonians – hence all the "minute" and "second" stuff because they used a base-60 numbering system. And all the calculating systems came together with development of geometry and trigonometry and the invention of the telescope with it's astronomical measurements.

Lewis and Clark's scheme to get government funding to go offroading fit in to Jefferson's vision for quantifying scientifically the "inventory" of his country, so was actually granted. With the proviso, however, that they: "take careful observations of latitude & longitude… (of) places & objects… of a durable kind… with great pains & accuracy, to be entered distinctly & intelligibly… (so that they could be read by) others as well as yourself."

Jefferson wanted to know what was in the "grocery bag" of his latest purchase – Louisianna. And he didn't want to take any chances, so told them to make "several copies" of their journals in their "leisure time," spread them out for guarding and put one of the copies on "paper of the birch… (as) less liable to injury from damp." [The free Google Earth map: Lewis and Clark]

Far better it is to dare mighty things, to win glorious triumphs even though checkered by failure, than to rank with those poor spirits who neither enjoy nor suffer much because they live in the gray twilight that knows neither victory nor defeat.”
Theodore Roosevelt
They were to behave themselves with the natives, avoid undue risk and use their "discretion" in "the degree of danger you may risk." To their honor their "tolerance" for risk ranks them with the Columbus's and Magellan's.

Unfortunately, for Jefferson and the country, the adventuresome duo magnificently "talked the talk AND walked the walk" but never "finished the paperwork." Many of their painstaking sightings could never be calculated to completion, and even to this day have been wasted. Partly because of observational inaccuracies, partly from the death of Lewis and partly due to the horrendously difficult mathematical calculations which even the best mathematicians could/would not do.

Latitude - Simple

Compared to how I got the coordinates for the Wave, those two adventurers must certainly be considered to have "gone commmando" in determining their coordinates; longitude more than latitude.

Latitude, even for the Babylonians (if anyone bothered) is the easiest to derive because it takes advantage of a phenominal coincidence, a star exactly over the north pole, and can be calculated  by measuring one simple angle: Polaris over the horizon.

You see at the North Pole (90° latitude) the North Star is directly overhead, and all during the night the stars rotate in a circle around it. At the Equator (0° latitude), it is directly ON the horizon and the stars arc across the sky.

The very definition of Latitude was set up so that the degrees equal the angle of Polaris with little effort to calculate. The sun, moon or certain planets and starts could/can also be used by observing them on known days and reading latitude directly from tables (ephemerides or almanacs).

For Latitude Lewis' main problems were the apparent changes in the "real" horizon with terrain, erroneous angles due to light refraction in the atmosphere, estimating the center of the sun or moon when it was so big it over-filled your lens, and the fact that nowhere on the earth (except at 0 and 90° Lat) does straight down "really" point at the center of the earth – the place where all the tables and charts expect it to be – a problem called parallax.

All-in-all however, the explorers Latitude figures were accurate to within a fraction of a degree. An ancient Anasazi Shaman looking up at the stars while lying on his bed mat under the Falling Man glyph that he had etched earlier in the day, would find Polaris 36.507938° above the horizon.

The angle that Polaris is above the horizon even works when you are on a ship in the middle of the ocean (if you are in the right hemisphere), even better because you don't have pesky mountains to get in the way. The only problem there, is measuring an angle in a wildly rocking boat.

And, as a clever sailor, you also notice that all stars (except Polaris) move from east to west thereby crossing a carefully drawn line from due North to due South known as a meridian.

In fact even on a bouncing ship you can time when the star crosses (called the transit) – if, you can just keep sight of North and South (we don't worry about what we don't know about, like Magnetic North fluctuating).

Longitude - Not So Much!

We've got North-South position down pat (sort of), but Longitude – now that's an equine of an alternate hue. Unfortunately we don't have anything like a geosyncronous Polaris over the equator but we do rely on another very useful trick: the Earth rotates; and being 5.9742 × 10²¹ metric tons it does it very, very steadily.

Longitude describes the East-West position of a place on the Earth's surface. As such, you've got to either have some way of measuring your unknown position in relation to some known position on the earth; OR, some fixed points "way out there" (like Polaris) whose observations vary some way in relation to longitude.

As it turns out, mariners had the later before they had the former. By 1610 Galileo had devised a way to determine longitude by observation of the passage of the four moons of Jupiter.

In the early 1800's, approximations of longitude were made by measuring the changing angular distance between the Moon and a prominent star such as Antares, doing calculations and comparing with charts. However, several sequential, meticulous measurements followed by extensive calculations were required – almost impossible in the field.

We know that those stars overhead travel in an arc to cross that meridian every 24 hours (360°). Which means that they must travel through an east-to-west 15° arc in one hour [ain't base 60 grand?].

Soooo… if only we had a known, fixed point on the earth which we could reference; AND, then knew what time it was THERE when we saw the sun transit OUR meridian THEN we could calculate what degree of longitude our point was at by the simple difference.

It took until 1735 for someone with money to get pissed enough at the whole mess to figure that "someone out there has got to be smart enough to figure this out." The British Government offered a 20,000£ (almost 49 million USD today based on avg earnings) reward to anyone who could figure it out.

A clock-maker, John Harrison, devised a working spring-based chronometer to win the prize; but, it took another 38 years (1773) to get one working and durable enough for extensive field observations. For Britian's efforts in "flashin' the cash", they got to be "Prime Meridian" – ZERO Degrees!

Lewis and Clark carried this version of Harrison's chronometer with them. Their difficulties with accuracy in measuring Longitude included ALL those for latitude PLUS the fact that their watches kept running down!

We've already said that 15 degrees of longitude is the equivalent to a time difference of one hour. So an offroader found out how far east or west he was from home (or Greenwich England), by meticulously watching the sun cross the meridian straight up (noon) then seeing what the exact time was back home. From there you look at charts and do math.

If your chronometer stopped, you could re-calibrate it… yep, you guessed it, by yet more astronomical observations, and charts! It was a constant tap dance from taking sightings to correcting the clock to correcting the compass to taking sightings to… Never ending.

For the most part, Lewis and Clark came home with a bunch of recorded numbers which had never been calculated… and still never have.

But YOU can do something that they couldn't. Next time you are out at 21 Goats Petroglyphs, pull out the lawn chair, slip on the protective eye gear and watch the sun transit the meridian. At that precise moment, pull out your digital watch (which you've conveniently set to GMT) and what do you think it will read? … yes, 7:36:45.836!

The Goats are at: 36.501668°; -114.190984° [N36° 30' 6.005", W114° 11' 27.542"]. Meaning 114.190984° in longitude behind Greenwich England in earth roation. Moving at 15° per hour, (i.e. ÷ by 15) the Prime Meridian is thus 7 hrs 45 minutes and 45.836 seconds of roation ahead of 21 Goats.

Ahhh… we'll never be lost again!

Learn A Little More

The needle of a magnetic compass points to True North only along a few ever-changing lines on the earth. It turns out that what we measure as the earth's magnetism is generated by all the bubbling/flowing magma at the center of the earth which is constantly (but slowly) changing.

A survey in the continental United States, based on Magnetic North, currently can depart from true north by as much as 15° West to as much as 20° East. This range of values was about the same for 1805.

To correct a survey based on Magnetic North, the magnetic declination (or variation of the needle) must be known. The direction of True North can be found by several methods:

In far northern latitudes during the winter the bearing of the sun can be taken at Local Apparent Noon. Another method is to take the bearing of the sun, its altitude and the chronometer time of the observation. Still another method is to observe the bearing of Polaris and the chronometer time of the observation.


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