[Proj] Comments of tmerc, etmerc and ftmerc errors

Clifford J Mugnier cjmce at lsu.edu
Thu Jun 12 14:43:57 EDT 2008

I recommend that error contour maps ( showing the graticule only), be posted to the PROJ4 documentation site - perhaps as "pdf" files.  This would be a superb graphic of "why this and why not that."  Similar things were immortalized in some of the German language geometric geodesy texts of the 1940s - 1950s, but never translated or duplicated into British or American English texts.  
Furthermore, I would like to suggest that a special routine be developed in PROJ4 for "SAFETY FAIRWAYS" that can accommodate the United States Code of Federal Regulations, Volume 33, Chapter 1, Part 166 (33 CFR Ch.1 Pt. 166, Subparts A & B) per the authority of 33 U.S.C. 1223; 49 CFR 1.46.  
This covers the concept of what the United States government has established as international guarantees that certain shipping lanes within our Exclusive Economic Zone are free of hazards to navigation.  Established by the U.S. Army Corps of Engineers and enforced by the U.S. Coast Guard, this affects the Oil Exploration and Production Industry within U.S. Waters, primarily in the Gulf of Mexico.  The same concept applies internationally to "TRAFFIC SEPARATION SCHEMES" as published by the British Admiralty Office, UK. for the rest of the world.
When prospecting for hydrocarbons in offshore areas, national governments award exploration and production concessions or ("Lease Blocks in U.S. waters), to Oil Companies according to various national or provincial Grid systems such as UTM, Lambert Conformal Conic (both fully conformal conic as well as the French Army Truncated Cubic), Cassini-Soldner, Ellipsoidal Bonne (really!), Oblique Stereographic, Transverse Mercator, etc.  In the Gulf of Mexico, the State Plane Coordinate Systems of Louisiana, Texas, and the UTM are used by the U.S. Bureau of Land Management through the administration of the U.S. Geological Survey, Minerals Management Service for putting out for public bid leases for Offshore Lease Blocks.  These concession leases are defined by a particular grid system, but a senior right exists for any and all Safety Fairways and/or Anchorages as published in 33 CFR, Ch.1, Pt.1.  Interestingly, these Safety Fairways (as well as the international Traffic Separation Schemes), are defined by end points given in latitude and longitude and represent ellipsoidal loxodromes (rhumb lines) in between the end points !!!  The geodetic/cartographic conundrum facing the GIS Specialist is, " where does a particular ellipsoidal loxodrome intersect a specific Grid line defining a Lease Block?"
The conundrum exists because a Lease Block Concession Grid line is only a straight line in the projection space, (such as UTM, Lambert Conformal Conic (both fully conformal conic as well as the French Army Truncated Cubic), Cassini-Soldner, Ellipsoidal Bonne (really!), Oblique Stereographic, Transverse Mercator, etc.), but in that same projection space the LOXODROME is curved!  Likewise when the loxodrome is a straight line in isometric normal Mercator space, ... , all other Grid lines are curved!

The solution is iterative.  The loxodrome is easily defined in isometric normal Mercator space by using the two end points as arguments to simply form the equation of the first line, (y = mx + b)1, find its approximate intersection with a particular Grid line by taking two points that define an enclosing section of that Grid line, (enclosing the loxodrome), take the inverse of those two Grid points to obtain the latitude and longitude, re-compute them into isometric normal Mercator space, form the equation of that second line, (y = mx + b)2, and then solve for the intersection of the two lines.  Then continue to iterate on smaller and smaller sections of the Grid line until one finds the exact intersection point. 


How important is that?  Some years ago, one Oil company to remain unnamed - but still in business nowadays, won a Lease Block bid that was due South of Houston in the Gulf of Mexico.  They filed an application for permit to drill from the Corps of Engineers and were denied a permit to drill in their own Lease Block because of the presence of a protected Anchorage (defined by loxodromes)!!!  They sued the Federal Government, the Minerals Management Service, the Corps of Engineers, the Coast Guard, etc., and LOST.  The Federal Judge told them that it was their responsibility to check to see if any protected waters according to 33 CFR, Ch.1, Pt.1. , encroached on any Lease Blocks they were interested in.  It cost that Oil company many, many millions of dollars for that little "geodetic/cartographic conundrum." 


Currently, all offshore exploration and production surveying done is with GPS.  The National Datum of the United States is the North American Datum of 1983, identical to the WGS84 Datum at the one-meter level.  However, all Lease Blocks in the Gulf of Mexico are still defined on the North American Datum of 1927!  Perplexing, isn't it?


When one looks at current Oil Industry-oriented maps of the entire Gulf of Mexico, there's four different map projections exhibited.  The UTM lease blocks in Federal waters are straight only on the Transverse Mercator, the two State lease block areas are straight only  on Lambert (Texas and Louisiana) zones , and the Safety Fairways are straight only on the normal Mercator.  Funny thing ... everything is forced to be a straight line ... a mathematical impossibility.


Such are some of the geodetic/cartographic tribulations of today's GIS Specialists in the Oil Patch.  Similar conundrums are found elsewhere in the world in association with Traffic Separation Schemes.  It seems to me that this would be a nifty subroutine package for PROJ4 with respect to practical applications.


There's my two cents' worth for the day.


Cliff Mugnier



From: proj-bounces at lists.maptools.org on behalf of Gerald I. Evenden
Sent: Thu 12-Jun-08 11:17
To: PROJ.4 and general Projections Discussions
Subject: [Proj] Comments of tmerc, etmerc and ftmerc errors

Two testing grids were made on tmerc, etmerc and ftmerc: in a geographic range
of 0 to 6 degrees longitude by 0 to 84 degrees latitude and 0 to 500km
Easting to 9,700km Northing.  The most apparent problem is that for
reasonable demonstration the results should be presented as contour maps of
the errors but inclusion of same is not possible in this email list.

A simple summary can be made: in the lat-lon tests tmerc maintained an
accuracy of <0.7mm (as would be expected occurred at the equator) and
comparison of etmerc and ftmerc was always less than 1micron.  In the
Easting/Northing test, tmerc failed dismally at 21 meters at about 58degrees
longitude/84.78 degree latitude.  Again, ftmerc and etmerc maintained
agreement to within 1micron.

I believe it is quite safe to say that tmerc remains the projection of choice
for UTM applications: adequate accuracy performance and best computational
speed. Note that the lat/lon test was for 12 degree zones and thus tmerc is
would appear suitable for what I understand of the 10 degree zones used for
some Canadian province grids.

As for extended range, I will probably check the agreement of etmerc and
ftmerc over extended range but this is of lesser priority.  Extended
geographic range usage of any conformal projection is a contentious issue as
any resultant grid system has sufficiently large scale errors as to make the
Cartesian usage of the grid very questionable.  Usage of conformal maps for
global mapping is very questionable because of the distortion.  Normal
Mercator persists only because of its usefulness in rhumbline navigation
which is probably fading due to GPS.  Lastly, equal-area mapping is far
preferable for thematic mapping as it at least give a better perspective of
the area extent of regions that is, in most cases, more important that an
often poor preservation of regional shape.
The whole religious complexion of the modern world is due
to the absence from Jerusalem of a lunatic asylum.
-- Havelock Ellis (1859-1939)  British psychologist
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