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# Why Ngs Publishes 5-decimal Place Positions

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NGS has been publishing the precision of coordinates in arc seconds to 5 decimal places since the early 1970's, many years prior to the adjustment of NAD 83. Users of coordinate data are often cautioned to NEVER look at the number of digits to the right of the decimal point as an indicator of positional accuracy. This data element is always linked on the NGS data sheet with one the 7 horizontal orders of accuracy “HORZ ORDER,” adopted by the Federal Geographic Data Committee:

A-Order (1:10,000,000)

B-Order (1:1,000,000)

First-Order (1:100,000)

Second-Order Class I (1:50,000)

Second –Order Class II (1:20,000)

Third-Order Class I (1:10,000)

Third-Order Class II (1:5,000)

Many years ago, NGS adopted 5 decimal places in arc seconds to be able to realize the spatial integrity of points with well determined horizontal positions to the millimeter level. This certainly doesn’t imply that all points are known to mm accuracy, but it maintains the relationships between points at mm level. For example, look at the data for stations ELK LICK (JW1482) and ELK LICK 2 (JW1480). Note that the distance between them was measured with a very accurate invar steel tape as 55.146 m and is well known at the mm level. If you compute the distance between them by using the program INVERSE from the NGS Geodetic Tool Kit, you will find that the distance returned is 55.146 m. If however you truncate their respective positions to say three decimal places, which might more closely represent their actual positional accuracy, the distance returned by INVERSE is 55.130 m, or a difference of 1.6 cm. This may not seem like much, but as you add observations across the country these errors would accumulate very quickly to an unacceptable level.

Surveyors and other spatial data users are constantly being trained by NGS at workshops and seminars on the intricacies of how to properly use and evaluate the accuracy of geospatial data.

So the question becomes just how accurate can a long distance measurement be? In a modern machine shop with very new and accurate equipment, 5 decimal places to the right (of an inch) can be made accurately by only the most expensivally geared facility such as in the Aerospace industry. 7 decimals to the right (of an inch) is virtually unheard of. And here, we are talking objects that are measured in inches or maybe a few feet.

I still find it difficult to believe that surveyors using 'one stick at a time' were capable of 5 or 7 decimal places to the right (of a foot) and be that accurate over many miles, 50 or 100 years ago.

(ohhhhh....I know.....I probably just stepped on the toes of 40 or more people on this forum!)

The measurement of distances, often called baselines in the days of triangulation could be measured to a precision of 0.1 mm with a 50m properly calibration, aligned and oriented invar steel tape. Baselines in lengths of 8-15 miles were often measured with tapes and bars (prior to tapes in the early 1900s) to precisions of 1:1,000,000 or 1 cm in 10 km miles. These measurements were quite common in the geodetic community. The typical land surveyor with an uncalibrated steel tape could typically achieve 1:10,000 or 1 ft in 1.9 miles and with more advanced electronic devices using laser, infrared or microwave precisions in the range of 1:30,000 to 1:500,000 were not at all uncommon. Today with GPS, using the appropriate equipment and techniques, it is not uncommon to achieve 1 cm accuracy over 800 - 1000 km.

Like Spoo, I am having some trouble with this and have posted in the NGS forum in the Topozone topic where this originally came up.

Baselines in lengths of 8-15 miles were often measured with tapes and bars (prior to tapes in the early 1900s) to precisions of 1:1,000,000 or 1 cm in 10 km miles.

1 cm in 10 Km is indeed an impressive accuracy rate if you are refering to the "one stick at a time" method used 50 or 100 years ago!

Does this mean that a stack up of error could have been possible of 50 cm as measured 500 Km inland from the shore? It seems that if an error of 1cm/10Km from point A to point B or B to C, etc, is excellent and acceptable. But doesn't that mean that from point A to pont Z the accuracy drops tremendously?

(Sorry if I am beating a dead horse here.)

If all distances are the same and all measurements are to the same accuracy, the rate of random error propagation is the square root of the number of legs (sticks) times the error level (an ellipsoid) in one leg. In other words, if one leg has an error level of 1 cm, then 25 legs has an error level of 5 cm.

An adjustment distributes the 5 cm as mathematically fairly as possible among the 25 legs measured, but can't reduce it.

To Spoo's point:

Sure, if you chained, ran rods or otherwise measured for 500 KM from a known point (a point assumed to be known with a high degree of accuracy), you would "stack up" errors as described by the BDT.

But since about 1785 (when both latitude and longitude could be determined with relative ease and accuracy),what surveyor would go more than a few miles without updating his or her position by some other method than by linear measurement? That is, the problem of "stacking up" errors while measuring from Point A to Point Z would not be the main problem of ascertaining the accuracy of the position at Point Z because Points D, G, K, N, R and U would be re-surveyed along the way. The calculated deviation between A and D would be spread among A, B, C and D, and then Point E would be measured as accurately as if it were Point B. So, while the error over 500 KM would potentially be large, the actual error would be limited to that expected for a measured distance of, say, 20 Km.

At least that how I think it would work.

I guess it would be good to know how long baselines typically were in the "old days". FWIW, Mason and Dixon updated their latitude position about every 11 miles along the 238-mile PA-MD boundary line.

W

Edited by seventhings

seventhings -

How did they update their latitude? Astronomy?

DaveD, BDT and seventhings:

Thank you for your patience with me. I think I am beginning to get the drift of this. You have to understand that my idea of Lat and Long was back when I was flying Helos in the early-mid 70's. +/- a few seconds of arc was usually adequite to get us back home or to a known location.

One last question: Are you folks saying that an known error (to make up numbers) of 5cm across 10 surveyed points is now averaged across them all?

Thanks again.

How did they update their latitude?  Astronomy?

Yes. Astronomy was a part of schooling and use by past surveyors and still is, is some degree, in use by modern surveyors. The celestial bodies easily located the latitudinal position of any point on the ground (or in the ocean), but both the sun and the North Star were used in determining the bearings of line on the ground as well. Modern geodesy courses taught in colleges are full of astronomic theory and practice which teach those principles. Many State's Land Surveyor's registration exams do have questions about astronomic calculations.

With the advent of GPS, astronomy, along with most other old school surveying techniques, is no longer used in practice, but the theory behind it still is.

- Kewaneh

seventhings -

How did they update their latitude? Astronomy?

Yes, the stars were used. Also, Triangulation, Trigonometry and Least Squares Adjustments.

Not all measurements were purely Linear. They were Optically and Mathematically cross compared to many other observations many times, many ways, to remove cumulative errors.

Rob

BDT -

Astronomy? Yes, that's how they did the latitude. Charles Mason was an astronomer and mathemetician. Jeremiah Dixon was a land surveyor.

W

I guess it would be good to know how long baselines typically were in the "old days".  FWIW, Mason and Dixon updated their latitude position about every 11 miles along the 238-mile PA-MD boundary line.

The base line is measured carefully, since the accuracy of the whole triangulation network depends on it. Astronomic observations ("Laplace stations") are done periodically to control the accuracy of azimuths and adjust for the deviation of the vertical.

In Bomford's classic Geodesy, he recommended "probably ... at least 3 miles, and generally 6 or more" for a base line. He stated that the error should not be more than 1:1,000,000 (1 cm over 10 km). He was writing in 1952, and complained that in "old work [the error] may be serious."

By comparison, he noted that "the probable error of bases measured in India with bimetallic bars between 1835 and 1870" was equivalent to 2.6 cm over 10 km.

The Borden Base Line in Massachusetts is supposedly the first triangulation base line in America and was completed in 1831. It was 7.42 miles and was measured with rods and microscopes! The Mason Dixon line wasn't a triangulation network, it was based on astronomical tables and direct measurements.

Spoo -

On your last question - imagine surveying around the perimeter of a small town or something. Station 1 is a fencepost. You survey 99 lengths of tape (sticks, legs, whateveryoucallit). Station 100 is also the original fencepost, so station 1 = station 100. You put all the measurements into a computer, and get it to do all the trigonometry required to get coordinates for each station relative to those of station 1, which you could use 0 0 0 as its coordinates for this example. These could be called 'raw coordinates'.

Anyway, to your dismay, you find that the coordinates of station 100 are NOT 0 0 0. Instead they are some distance, say, 3 feet, away.

You then submit all the measurements to an adjustment program. It adjusts (re-calculates) the coordinates of all the stations between station 2 and station 99 using least squares (or perhaps some other mathematics) while using some weighting rules you give it, such as "longer tapings should get a larger share of the distributed error". The adjustment program also makes sure that station 1 and station 100 both have coordinates 0 0 0 (or perhaps some real coordinates if the fencepost has a PID). This equivalence (station 1 = station 100) is an example of a constraint that you can give an adjustment program. Anyway, the 3 foot error you noted gets distributed among the stations so that the loop 'closes'.

Did the overall accuracy of the survey increase because of the adjustment? No.

If you do the entire survey all over again and adjust it, will you get the same coordinates as te first time? No.

You will probably be again adjusting an error of around 3 feet, likely in a different bearing.

The first high quality geodetic triangulation baselines in the U.S. were observed under the direction of Ferdinan Hassler, first superintendent of the Survey of the Coast in 1817 in support of his work in the New York city area. The baselines, referred to as the English Neighborhood and Fire Island baselines were 9.4 km and 7.8 km long respectively. Details of this work are provided in "Principal Documents Relating To the Survey Of the Coast Since 1816" published by Hassler in 1834.

Holograph,

My3617 (Beaconpole-Borden 1834) is the oldest benchmark in R.I. and MZ1913 (Jilson Borden) 1836 is the oldest in VT. Do you know if they have anything to do with the person who setup the Borden baseline in Massachusetts.

Dave

Edited by ddnutzy

It's very likely that those stations were established by Simeon Borden who was in charge of the Massachusetts Commonwealth Survey from 1834-1838.

FZ0911                          STATION DESCRIPTION

FZ0911

FZ0911'DESCRIBED BY COAST AND GEODETIC SURVEY 1933

FZ0911'9.5 MI SE FROM BLOWING ROCK.

FZ0911'IN CALDWELL COUNTY, 9.5 MILES SOUTHEAST ALONG U.S. HIGHWAY 321

FZ0911'FROM BLOWING ROCK, WATAUGA COUNTY, 100 YARDS NORTH OF THE HOME OF

FZ0911'DISTRICT FIRE WARDEN LEE CARLETON (IN 1934), 50 FEET EAST OF THE

FZ0911'CENTERLINE OF THE HIGHWAY, IN THE TOP OF A ROCK IN THE SIDE OF A

FZ0911'HILL ACROSS A STREAM FROM THE HIGHWAY, AND 5 FEET ABOVE THE LEVEL

FZ0911'OF THE HIGHWAY.  A STANDARD DISK, STAMPED 1920.462 L 48 1933.

FZ0911

FZ0911                          STATION RECOVERY (1998)

FZ0911

FZ0911'RECOVERY NOTE BY NORTH CAROLINA GEODETIC SURVEY 1998 (MDB)

FZ0911

FZ0911                          STATION RECOVERY (2003)

FZ0911

FZ0911'RECOVERY NOTE BY NATIONAL GEODETIC SURVEY 2003 (JGE)

FZ0911'HANDHELD GPS POSITION = 36.055564256385, -81.6041170712561

That does seem to be a rather cavalier level of confidence.

Photographs?????

Wrong photographs, at that. Oh well. Must be experimental.

BDT:

Actually, NC photos generally are accurate. I've met the submitter. Next trip to the NCGS office, I'll ask him what happened.

Better yet, it is a lovely day. Perhaps I'll dash up to Blowing Rock and re-shoot the photos.

By the way, there is a lovely waterfall on the same road. I've been there, but it was in my pre-benchmark-hunting days. Until I read this description, I did not realize that this narrow twisting road used to be the US highway.

The things we learn in this hobby......!

Paul

[Edit: typo. Grrrr]

Edited by PFF

PFF -

I'm wondering whether any of the photos from GEOCAC logs are being included that way, or was FZ0911 just an experiment? I have never seen this before.

Here is another Borden benchmark, unfortunately reported gone. It sits on top of Blue Hill just outside Boston. The triangular plate over the spot has a nice description.

A quick search of the interweb (as my son calls it) turned up this article:

Seabrook/Salisbury monument I took a quick look to see if I could find it here and discovered this: Salisbury Marsh Monument.

Just answered my own question. This biographical sketch found says that he invented "an apparatus for measuring the base line of the trigonometrically survey of Massachusetts, which was found to be more accurate and convenient than any instrument of the kind then in existence."

http://www.famousamericans.net/simeonborden/

OK.....just to irratate every one once more......

If the above example by BDT is accurate, that is, if a 100 point survey leads to a 3 foot error, and if that error is repeatable, then the accuracy CANNOT be +/- 7 decimal places.

I am sorry......I have tried to explain that 7 decimal places to the right of an inch is virtually IMPOSSIBLE in the world of machining. I just find myself in disbelief that 7 decimal places to the right of a FOOT can be held on the face of this planet.

Other evidence and spokes people here have indicated that a measurement to a single location from several different points would give different relationships and numbers. I can accept and understand this. For instance, if someone starting from a known point in New York, measured to a point in Kansas, the co-ordinates would be different from a person that measured TO the same place but started from California. Both would be correct. Neither, unto itself, would be wrong.

I therefore see it as a matter of RELATIVITY. Seven decimal places can only be relevant to a particular observation but not to ALL observations.

signed, He of Little Faith

If the above example by BDT is accurate
I invented those numbers out of thin air. They are in no way meant to represent actual values in NGS surveys.

BDT.....my apologies.

Bad choice of words on my part. I really did understand that you were giving an exagerated example for my benefit and NOT citing an actual survey.

I think my point of relativity still sticks. NO SURVEYOR has been wrong with his published results, it simply is a matter of where one is measuring from.

I'm wondering whether any of the photos from GEOCAC logs are being included that way, or was FZ0911 just an experiment? I have never seen this before.

Newer marks (DG4268, DG4269), set by engineering firms, tend to have photos.

But here's one that stands out from all the rest: DE5174.

We've had this feature for a while. I've not had time to put any of my photos into the system. Most of the time, the description is adequate and I don't think the time spent formatting photos justifies the value. However, one project for this winter is to go back and add some pictures, especially where it may be of use.

Again, peek at DE5174 before logging off.

-Paul-

So, I guess only the original monumenting agency can get photographs into the database where it can be retrieved through a datasheet. Recovery photographs ain't there yet, it seems.

I am sorry......I have tried to explain that 7 decimal places to the right of an inch is virtually IMPOSSIBLE in the world of machining. I just find myself in disbelief that 7 decimal places to the right of a FOOT can be held on the face of this planet.

I'm not sure where you're getting the "7 decimal places to the right of a foot" from. The accuracy is measured as a proportion, so an error of 1:1,000,000 means 1 cm over a 10 km line.

The theory of errors offers insight into the nature of cumulative random errors. Suppose a 10 km baseline was measured in 50 m. segments, so there are 200 legs. The probable error of the sum of the 200 measurements is about sqrt(200), or 14 times, the probable error of each measurement, if you assume that the errors are random and equally likely to be larger or smaller than the true value. That means that each 50 meter leg needs to have a probable error of less than 0.7 mm.

That was doable. In fact, it was possible to routinely measure each leg to 0.2 or 0.3 mm. Each leg was measured several times and the results were averaged. The probable error of the average of multiple observations of the same quantity is roughly 1/sqrt(n) times the error of one observation, so if each leg was measured 5 times, the probable error of each leg would be about 0.15 mm. That's comfortably within the error required to obtain 1:1,000,000 accuracy over the 10 km line.

Of course, there could also be systematic errors, but those could be controlled by careful calibration and technique.

edit: typos, of course

Edited by holograph

BDT wrote:

So, I guess only the original monumenting agency can get photographs into the database where it can be retrieved through a datasheet. Recovery photographs ain't there yet, it seems.

In North Carolina, recovery photos can be submitted. See FZ2035 as an example.

-Paul-

Edited by PFF

I therefore see it as a matter of RELATIVITY. Seven decimal places can only be relevant to a particular observation but not to ALL observations.

ALL surveying is a matter of relativity, whether it is geodetic surveying, cadastral surveying, boundary surveying, construction surveying, running a level, or anything else. It all has a start point - a benchmark of some sort - and all measurements are relative to that start point. In a triangulation network, whether GPS or more traditional survey methods are used, all of the points are measured relative to each other. Essentially, each point is the other point’s benchmark, and in most cases, a point may have multiple benchmarks that it is being tied to. In order to properly define a location of a given point relative to other multiple locations, 4, 5, 6, & 7 decimal places may be necessary to accurately define that location of that point.

These decimals places are not just made up - they are calculated - sometimes painstakingly calculated. The data from the measurements is evaluated along with the measurements themselves. If GPS was used, the satellite configuration along with the time span of the observations are also evaluated and added to the mix. GPS post processing software, along with a good portion of a calculative process called ‘least squares adjusting’, determines the most probable location (not necessarily the exact location) of a given point relative to the others in the network. Error factors (a plus/minus factor) to that point, are also calculated. When the most probable locations are used as definite, exact locations, and simple math is used to determine the distance (horizontal or vertical) between them, it is very possible to get an unbelievable number of digits behind the decimal point. These digits may not be practical, only theoretical, but they do serve a practical purpose in all observations, as described by DaveD in the third paragraph of his original post.

This thread is somewhat bifurcated and I'd like to get that sewed up (being the main guilty party myself).

On the sub-topic of recovery photos being incorporated/not-yet-incorporated by the NGS, ArtMan has created an excellent topic in the NGS FORUM asking the NGS 5 questions regarding the status of uploading recovery photographs and incorporating them in datasheets.

I haven't posted there; waiting for the NGS to respond.

PFF has given us evidence that some recovery agencies' (but no evidence of GEOCAC examples) recovery photographs have been incorporated (with a link to the photos) into NGS datasheets.

In my opinion, any further discussion of the important topic on the NGS' use of recovery photographs should be over on the NGS Forum side, leaving this topic to the original discussion of the 5 decimal points.

In regard to the 5 decimal points topic, I agree with Kewaneh's post. But let me try to put it into a perspective with Spoo's points.

There are really 3 realities (or sets) of data we are dealing with in an adjusted survey net (group of stations).

1. The actual/real/exact-coordinates-forever-unknown/infinite-decimal-accuracy positions of the survey points.

2. The observed/measured/calculated coordinates of the survey points without/before adjustment from the perspective of the first survey point (origin).

3. The adjusted coordinates of the survey points.

Spoo's perspective, I believe, is of reality #2, and he is correct - one cannot/should-not report any more precision than the accuracy that the process is determined to have, and especially no more precision than the actual measuring precision of the measuring equipment.

For a moment let's explore the caveat I just mentioned, using a ghastly example. Imagine that one measurement (a distance or an angle or a GPS position) within the survey net was waaaay off - some nasty transcription error or something happened. Normally the adjustment program will stop (or at least give you a very very bad report) when it sees such a thing and tell you to start over.

But, lets imagine you tell it to run anyway to achieve its usual goal of many-decimal accuracy and it does that. When it finishes, from its perspective, the relationships between the adjusted coordinates will be accurate to many decimal points. Their differences from reality #1 will of course all be several meters off! (And the adjustment program would be saying - don't blame me - I told you it would be no good. )

So I'm imagining that the NGS position is:

we verifiy the accuracy of the survey measurements in establishing these stations and have made sure that our adjustment program didn't give us a bad report on them, and within that framework, we're saying that using 5 decimal places is a proper reflection of the combination of of our high-precision adjustement of excellent data.

Another important point was made by holograph and another way of putting it is that fortunately errors don't just plain add up over a sucession of survey legs; instead they add up in a square-root fashion. This is a slower rate of addition! (25 measurements have only 5 times as much error as one measurement.) The reason is this: errors can and will be both plus-and-minus along any given bearing. The first leg might be measured longer than it really is, while the second leg might be measured shorter than it really is, etc. This is an important thing to keep in mind when looking at, and estimating, error accumulation in a series measurement processs like surveying.

Regarding FZ0911 it appears the photograpghs were by NGS and I couldn't see where they were the wrong photo's. The coordinates on the old benchmarks are scaled from the USGS quad sheet unless it's been positioned horizonally. There is an effort underway to collect mapping grade coordinates on found benchmarks and to include it in the text of the recoveries to aid in future recovery. I'd agree that listing the coordinates out to as many places as are shown might lead someone who didn't read the"handheld gps position" statement first, to think they were better than they are. Perhaps a disclaimer of some sort might be in order.

I'd also venture to bet that the photographs of FZ2035 are those submitted by the height modernization contractor "Geometrics GPS Inc." and not the geocacher.

amberlu -

Yep, it appears that they fixed the photograph links for FZ0911; they are correct now.

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