Wolf and Ghilani "Adjust" Mark to Mark Distance

Wolf and Ghilani "Adjust" What Textbook Version?

And what software version?

From what I recall the mark to mark distance is along a complex curve.

Paul in PA

A mark-to-mark distance is a chord distance, from the mark at one end to the mark at the other end. This is what GPS software provides when it gives a DX, DY, DZ vector, those are also mark-to-mark values, but with more information thrown in (i.e. 3-D instead of a line with only magnitude). It can be converted to an arc distance, but the difference only becomes noticeable for LONG lines, mostly lines longer than can be measured with most modern EDM's. In the old days the microwave and big lightwave EDM's could measure many kms, but most (all?) instruments sold now have a range of only a few km. Of course with GPS vectors there can definitely be a difference between the chord and the arc.

To me, mark-to-mark is the most natural way to store a distance (and also a zenith distance). I transform all of my observed data to this before storing in a database. It also makes it very easy to combine with GNSS vectors, or to compare against GNSS vectors. A mark to mark distance is invariable, does not depend on datum or elevation. Between any two points there are an infinite number of "horizontal" distances and slope distances, but only one mark-to-mark distance. Similarly, the reduced ellipsoidal distance (which is an arc distance) depends on choice of ellipsoid, although if one uses a geocentric ellipsoid that problem is eliminated, so an ellipsoidal distance on a geocentric ellipsoid is also unambiguous.

The adjustment software that I use (Geolab) uses mark to mark, although it can also use a slope distance along with HI and HT, which it uses to reduce to the marks.

I am an advocate of using the ECEF system for data storage (i.e. positions and observations), but I will admit I am not there yet, as I need to modify some of my workflows and do some database programming before I can truly be there. Clients still usually want SPC, or UTM, although I did do some photo control projects this year and last where the client wanted ITRF XYZ along with XYZ standard deviations to control 3" (7.5 cm) imagery over very large areas.

Another advantage of using ECEF XYZ: doing an inverse, the mark to mark or chord distance is simply SQRT(X^2+Y^2+Z^2). So, if I have a reduced EDM mark-to-mark, it is a simple matter to compare against XYZ coordinates, no scale factor, no elevation factor.

In pure ECEF there is no ellipsoid, no curvature, just chords (straight lines) between points. It is a simple 3-D coordinate system with origin at the earth center.

Attorneys for Insurance companies contact me occasionally to compute distances in situations where someone has rented a vehicle and gotten into a crash where it is thought that they exceeded the maximum distance allowed by the rental contract. I obtain the Lat/Lon of the rental agency and of the crash site and compute the ellipsoidal geodesic on the GRS1980 ellipsoid using NGS (Inverse Geodesic) software and then write a letter documenting the procedure and the result.

Works in State and Federal Courts.

Wolf and Ghilani "Adjust" What Textbook Version?

6.1.0 - the most recent version

The software gives both mark to mark and ellipsoid distance between 2 lat/lon/h points. They are different, as expected. What I am wondering is, if I have ground coordinates for these points as well, should the inversed slope distance between them (assuming perfection), be the same as the mark to mark distance?

Having been tasked with expanding the coverage of a mine grid, I have lats and lons and local ground cords for several points, several km's apart.
I know there will be errors (grid is flat, earth is not, etc....) and would like a way to quantify the magnitude of the error.

Based on comparing inversed slope distances between the ground cords and the mark to mark distances from the lat/lons I'm getting about 85mm difference in a 5.8km baseline. That seems reasonable, but am I barking up the wrong tree?

Mark-to-mark distances are slope distances. To see this, look at the input to INVERS3D below. The lat/lon of the two stations are identical, but the ellipsoid heights differ by 100 meters. Note that INVERS3D returns 100 m as the mark-to-mark distance. Were mark-to-mark an ellipsoidal distance or a chord between two points on the ellipsoid, the mark-to-mark distance would be zero.

A bit more insight is available in Thaddeus Vincenty's short paper at this link:

http://www.tech.mtu.edu/courses/su3150/Reference%20Material/Vincenty.pdf

Output from INVERS3D

First Station : Sample Low
----------------
X = 897066.2735 m LAT = 36 0 0.00000 North
Y = -5087515.6482 m LON = 80 0 0.00000 West
Z = 3728191.6757 m EHT = 0.0000 Meters

Second Station : Sample High
----------------
X = 897080.3220 m LAT = 36 0 0.00000 North
Y = -5087595.3208 m LON = 80 0 0.00000 West
Z = 3728250.4543 m EHT = 100.0000 Meters

Forward azimuth FAZ = 0 0 0.0000 From North
Back azimuth BAZ = 0 0 0.0000 From North
Ellipsoidal distance S = 0.0000 m
Delta height dh = 100.0000 m
Mark-to-mark distance D = 100.0000 m

DX = 14.0484 m DN = 0.0000 m
DY = -79.6726 m DE = 0.0000 m
DZ = 58.7785 m DU = 100.0000 m

NGS HOME PAGE

Mark to Mark Chord Distance Should Match Traverse

I use the XYZ-XYZ as a first check against my traverse. Generally I just grab my calculator and don't bother with any adjustment program.

On any GPS project no mater what output is expected I carry elevations with my traverse. My side shots may never get elevations, but I know enough to not leave any observation on the table when I want quality results and be able to prove it.

Paul in PA

Thanks for the link - I thought Vicenty was some ancient Geodesist from the 15th century!

Mark to Mark Chord Distance Should Match Traverse

I seem to be on the right track then.

Mark to Mark Chord Distance Is A Ground Distance

I forgot to add that this check is directly against your ground field distance, no concern for SPC to grid conversions at that point.

Later on I compare my field traverse elevation differences to my GPS ortho elevations differences.

Paul in PA

Mark to Mark Chord Distance Should Match Traverse

Yes. The remaining question is what do you mean by ground coordinates? If they are of the SPCS scaled variety discussed elsewhere, then they are a type of plane coordinates. The differences you see may be the differences between slope distances and horizontal differences.

Paul is absolutely right in his post below. Mark-to-mark distances are ground distances; not horizontal, but ground.

Mark to Mark Chord Distance Should Match Traverse

The remaining question is what do you mean by ground coordinates?

They are derived from total station traversing in the mine grid - pure ground coordinates. There was 2km shots over a lake involved and other fun stuff.
The lats/lons came from a static network.

I created a localization file in the office with the 2 data sets and the residuals were so good, I wanted to find another way of testing the results....hence my Wolf and Ghilani textbook has been on the table all day and I've been bugging you guys...:-)

Mark to Mark Are Points On The Ground

Setup on one point and observe the other, redo the opposite direction. Truth be told you may have some intervening traverse points but he solved traverse vector mark to mark should equal the mark to mark GPS. Pay attention to the fact that the GPS antenna to antenna vector is not the mark to mark vector. GPS software realizes the most critical point are those that are fixed in the ground and not floating at some antenna height.

Your traverse vector is from a point on the ground o a point on the ground assuming that you recorded your instrument and rod heights as good as your antenna heights.

Mark to mark/ground to ground means exactly that, do not think grid or SPC or LDP, just hard measurable values.

Paul in PA

Mark to Mark Chord Distance Should Match Traverse

But - if you've traversed several km's, there ought to be a slight difference, shouldn't there? And wouldn't it be the error in the ground coords?
Isn't that why the State Plane was developed; because when the old guys traversed between widely spaced geodetic control points they were getting sloppy checks - not because they were sloppy but because the earth is curving?
Sorry to keep bugging...!

Traverse Several Kilometers ?

Your mark to mark distances will be in the fixed WYZ system.

Your traverse is following the surface of the earth. Traverse software accommodates for refraction, etc. The chord traverse distances will follow the XYZ GPS points as they follow the curve. As long as you match point to point keep on going.

What you cannot do is ever assume that your traverse is on a flat piece of paper. SPC was designed to accommodate that problem.

Paul in PA

Traverse Several Kilometers ?

I think that we're hitting all around the answer to your question without really providing a specific answer.

If your ground coordinates include good ellipsoidal heights and your inverse computations correctly use those heights, and your lat/lon values for the points are perfect, then your inversed ground coordinate distance should match mark-to-mark computed distance. Differences in the computations come from variances in one or more of the inputs or differences in computational methods.

The first method for reducing measured slope distances to ellipsoidal distances given by T. Vincenty at the link in a previous post is very instructive. A ground slope distance is first reduced to a ground horizontal distance by the Pythagorean Theorem. The horizontal ground distance is then reduced to a chord distance between the two points on the ellipsoid surface. Note that all three of these distances are straight lines. Vincenty's final computation adjusts the straight chord to a curved ellipsoidal distance by adding a curvature term.

Also, the chord distance does not need to be adjusted for curvature for lines under 10 km. That may no longer be true in today's measurement world.

Computationally, you're on the right track.

Thanks for info.
I didn't realize or forgot the NGS pgm gave Mark-to-Mark distance.
Could have used that a couple of times in recent past to simplify checks.